![Definition 16.2.1. Transmitter {Xi} covers its receiver yi in the reference time slot if
SINRi =
Fi
i /l(|Xi − yi|)
W + I1
i
≥ T , (16.2)
where the interference I1
i is the SN associated with e
Φ1, namely, I1
i =
P
Xj∈e
Φ1, j6=i
Fi
j /l(|Xj − yi|) and
where T is some SINR threshold.
The last condition might be required in practice for xi to be successfully received by yi due to the use of a
particular coding scheme associated with a given bit-rate (cf. Section 24.3.4). It is also called the non-outage
or the capture condition depending on the framework.
Remark: Later, in Section 16.2.3, we shall also consider adaptive coding schemes in which the appropriate
choice of coding scheme is selected for each observed SINR level, which allows one to obtain a bit-rate
close to that given by Shannon’s law for all such SINR.
In what follows we focus on the probability that this property holds true for the typical node of the
MANET, given it is a transmitter. This notion can be formalized using Palm theory for stationary marked
point processes (cf. Section 2.1.2 in Volume I).
Denote by δi the indicator that (16.2) holds, namely, that location yi is covered by transmitter Xi. We can
consider δi as a new mark of Xi. The marked point process e
Φ enriched with these marks is still stationary
(cf. Definition 2.1.4 in Volume I). However, in contrast to the original marks ei, yi, Fi, given the points of Φ,
the random variables {δi} are neither independent nor identically distributed. Indeed, the points of Φ lying
in dense clusters have a smaller probability of coverage than more isolated points due to interference; in
addition, because of the shot noise variables I1
i , the random variables {δi} are dependent.
By probability of coverage of a typical node given it is a transmitter, we understand
P0
{ δ0 = 1 | e0 = 1} = E0
[δ0 | e0 = 1],
where P0 is the Palm probability associated to the (marked) stationary point process e
Φ and where δ0 is the
mark of the point X0 = 0 a.s. located at the origin 0 under P0. This Palm probability P0 is derived from the
original (stationary) probability P by the following relation (cf. Definition 2.1.5 in Volume I)
P0
{ δ0 = 1 | e0 = 1} =
1
λ1|B|
E
X
i
δi1(Xi ∈ B)
,
where B is an arbitrary subset of the plane and |B| is its surface. Thus, knowing that λ1|B| is the expected
number of transmitters in B, the typical node coverage probability is the mean number of transmitters which
cover their receivers in any given window B in which we observe our MANET. Note that this mean is based
on a double averaging: a mathematical expectation – over all possible realizations of the MANET and, for
each realization, a spatial averaging – over all nodes in B.
If the underlying point process is ergodic (as it is the case for our i.m. Poisson p.p. Φ̃) the typical node
coverage probability can also be interpreted as a spatial average of the number of transmitters which cover
their receiver in almost every given realization of the MANET and large B (tending to the whole plane; cf.
Proposition 1.6.10 in Volume I).
For a stationary i.m. Poisson p.p., the probability P0 can easily be constructed using Slivnyak’s theorem
(cf. Theorem 1.4.5 in Volume I): under P0, the nodes of the Poisson MANET and their marks follow the
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![distribution e
Φ ∪ {(X0 = 0, e0, y0, F0)}, where e
Φ is the original stationary i.m. Poisson p.p. (i.e. that seen
under the original probability P) and (e0, y0, F0) is a new copy of the mark independent of everything else
and distributed like all other i.i.d. marks (ei, yi, Fi) of e
Φ under P. (cf. Remark 2.1.7 in Volume I).
Note that, under P0, the node at the origin (the typical node), is not necessarily a transmitter; e0 is equal
to 1 or 0 with probability p and 1 − p respectively.
Denote by pc(r, λ1, T) = E0[δ0 | e0 = 1] the probability of coverage of the typical MANET node given
it is a transmitter. It follows from the above construction that this probability only depends on the density
of effective transmitters λ1 = λp, on the distance r and on the SINR threshold T; it can be expressed using
three independent generic random variables F, I1, W by the following formula:
pc(r, λ1, T) = P0
{ F0
0 l(r)T(W + I1
0 ) | e0 = 1 } = P{ F ≥ Tl(r)(I1
+ W) } . (16.3)
Note also that this probability is equal to the one-point coverage probability p0(y0) in the GI
W+M/GI SINR
cell model of Section 5.3.1 in Volume I associated with the Poisson p.p. intensity λ1. (see the meaning
of this Kendall-like notation in Section 5.3 in Volume I). For this reason, our Aloha MANET model is of
the GI
W+M/GI type, where the GI in the numerator indicates a general distribution for the virtual power of
the signal F and where the M/GI in the denominator indicates that the SN interference is generated by a
Poisson pattern of interferers (M), with a general distribution (G) for their virtual powers. Special cases of
distributions marks are deterministic (D) and exponential (M). We recall that M/· denotes a SN model with
a Poisson point process.
In what follows, we often use the following explicit formula for the Laplace transform of the generic
shot-noise I1 =
P
Xj∈e
Φ1 Fj/l(|Xj|), which is valid in the Poisson p.p. case whenever the random variables
Fj are independent copies of the generic fading variable F (cf. Corollary 2.3.8 in Volume I):
LI1 (s) = E[e−I1s
] = exp
−λ12π
∞
Z
0
t
1 − LF (s/l(t))
dt
, (16.4)
where LF is the Laplace transform of F. This can be derived from the formula for the Laplace functional of
the Poisson p.p. (see Propositions 1.2.2 and 2.2.4 in Volume I).
16.2.2.1 Rayleigh Case
The next result bears on the Rayleigh fading case (F exponential with mean 1/µ). Using the independence
assumptions, it is easy to see that the right-hand side of (16.3) can be rewritten as
pc(r, λ) = pc(r, λ, T) = E
h
e−µ(Tl(r)(I1+W)
i
= LI1 (µTl(r))LW (µTl(r)) , (16.5)
where LW is the Laplace transform of W and where LI1 can be expressed as follows from (16.4) when
using the fact that F is exponential:
LI1 (s) = exp
−2πλ1
∞
Z
0
t
1 + µl(t)/s
dt
. (16.6)
Using this observation, one immediately obtains (cf. also Proposition 5.3.3 and Example 5.3.4 in Volume I):
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![Remark: Sufficient conditions for I1 to admit a density are given in Proposition 2.2.6 in Volume I. Roughly
speaking these conditions require that F be non-null and that the path-loss function l be not constant in any
interval. This is satisfied e.g. for the OPL 3 and OPL 2 model, but not for OPL 1 – see Example 23.1.3.
Concerning the square integrability of the density, which is equivalent to the integrability of |LI1 (is)|2
(see (Feller 1971, p. 510) and also (2.20 in Volume I)), using (16.4), one can easily check that it is satisfied
for the OPL 3 model provided P{F 0} 0. Moreover, under the same conditions |LI1 (is)| is integrable
(and so is |LI1 (is)|/|s| for large |s|).
Proof. (of Proposition 16.2.4) By the independence of I1 and W in (16.3), the second assumption of Propo-
sition 16.2.4 implies that I1 + W admits a density g(·) that is square integrable. The result then follows
from
pc(r, λ1, T) = P{ (I1
+ W)Tl(r) F } ,
by the Plancherel-Parseval theorem; see e.g. (Brémaud 2002, Th. C3.3, p.157)) and for more details Corol-
lary 12.2.2 in Volume I.
16.2.3 Shannon Throughput of a Typical Node
In Section 16.2.2 we assumed that a channel could be sustained if the SINR was above some fixed threshold
T, which corresponds to the case where some minimum bit rate is required (like in e.g. voice). In this
section we consider the situation where there is no minimal requirement on the bit rate T and where the
latter depends on the SINR through some Shannon like formula. This is possible with adaptive coding,
where a coding with a high bit-rate is used if the SINR is high, whereas a coding with a low bit-rate is used
in case of lower SINR. We adopt the following definition.
Definition 16.2.5. We define the (Shannon) throughput (bit-rate) of the channel from transmitter Xi to its
receiver yi to be
Ti = log(1 + SINRi) , (16.12)
where SINRi is as in Definition 16.2.1.
One can then ask about the throughput of a typical transmitter (or equivalently about the spatial average of
the rate obtained by the transmitters), namely
τ = τ(r, λ1) = E0
[T0 | e0 = 1] = E0
[log(1 + SINR0) | e0 = 1]
and also about its Laplace transform
LT (s) = E0
[e−sT0
| e0 = 1] = E0
[(1 + SINR0)−s
| e0 = 1] .
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![Let us now make the following simple observations:
τ(r, λ1) = E0
[log(1 + SINR0) | e0 = 1]
=
∞
Z
0
P0
{ log(1 + SINR0) t | e0 = 1 } dt
=
∞
Z
0
P0
{ SINR0 et
− 1 | e0 = 1 } dt
=
∞
Z
0
pc(r, λ1, et
− 1) dt =
∞
Z
0
pc(r, λ1, v)
v + 1
dv (16.13)
and similarly
LT (s) = E0
[(1 + SINR0)−s
| e0 = 1]
= 1 −
1
Z
0
pc(r, λ1, t−1/s
− 1) dt = 1 − s
∞
Z
0
pc(r, λ1, v)
(1 + v)1+s
dv , (16.14)
provided P{SINR0 = T} = 0 for all T ≥ 0, which is true e.g. when F admits a density (cf. (16.3)). This,
together with Propositions 16.2.2 and 16.2.4, lead to the following results:
Corollary 16.2.6. Under the assumptions of Proposition 16.2.2 (namely for the model with Rayleigh fad-
ing) and for the OPL 3 path-loss model (16.1)
τ =
β
2
∞
Z
0
e−λ1K(β)r2v v
β
2
−1
1 + v
β
2
LW
µ(Ar)β
vβ/2
dv (16.15)
and
LT (s) =
βs
2
∞
Z
0
1 − e−λ1K(β)r2v
LW
µ(Ar)β
vβ/2
v
β
2
−1
1 + v
β
2
1+s dv , (16.16)
where K(β) is defined in (16.9).
Corollary 16.2.7. Under the assumptions of Proposition 16.2.4 (namely in the model with general fading
F),
τ =
∞
Z
0
∞
Z
−∞
LI1 (2iπvsl(r)) LW (2iπvsl(r))
LF (−2iπs) − 1
2iπs(1 + v)
dsdv (16.17)
and
LT (s) = 1 − s
∞
Z
0
∞
Z
−∞
LI1 (2iπvsl(r)) LW (2iπvsl(r))
LF (−2iπs) − 1
2iπs(1 + v)1+s
dsdv . (16.18)
Here is a direct application of the last results.
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![• the scale invariance property of the Poisson p.p. (cf. Example 1.3.12 in Volume I),
• the power-law form of OPL 3,
• the fact that thermal noise was neglected.
Note however the following important limitations concerning this scaling. First, when λ → ∞, the nodes
are closer to each other and one may challenge the use of OPL 3 (the pole at the origin is not adequate for
representing the path loss on small distances). On the other hand, when λ → 0, the transmission distances
are very long; communications become noise limited and the assumption W = 0 may no longer be justified.
16.3 Spatial Performance Metrics
When trying to maximize the coverage probability pc(r, λ1, T) or the throughput τ(r, λ1), one obtains de-
generate maxima at r = 0. Assuming that our MANET features packets which have to reach some distant
destination nodes, a more meaningful optimization consists in maximizing some distance-based character-
istics. In the coverage scenario, we for instance consider the mean progress made in a typical transmission:
prog(r, λ1, T) = rE0
[δ0] = rpc(r, λ1, T) . (16.19)
Similarly, in the digital communication (or Shannon-throughput) scenario, we define the mean transport of
a typical transmission as
trans(r, λ1, T) = rE0
[T0] = rτ(r, λ1) . (16.20)
These characteristics might still not lead to pertinent optimizations of the MANET, as they are concerned
with one (typical) transmission. In particular, they are trivially maximized when p → 0, when transmissions
are very efficient but very rare in the network. In fact, we need some network (social) performance metrics.
Definition 16.3.1. We call
• (spatial) density of successful transmissions, dsuc, the mean number of successful transmissions
per surface unit;
• (spatial) density of progress, dprog, the mean number of meters progressed by all transmissions
taking place per surface unit;
• (spatial) density of throughput, dthrou, the mean throughput per surface unit;
• (spatial) density of transport, dtrans, the mean number of bit-meters transported per second and
per unit of surface.
The knowledge of pc(r, λ1, T) or τ(r, λ1) allows one to estimate these spatial network performance
metrics. The link between individual and social characteristics is guaranteed by Campbell’s formula (cf. (2.9
in Volume I)).
In what follows we make precise the meaning of the characteristics proposed in Definition 16.3.1 for
Poisson bipolar MANETs.
The density of successful transmissions can formally be seen as the mean number of successful trans-
missions in some arbitrary subset B of the plane:
dsuc(r, λ1, T) =
1
|B|
E
X
i
eiδi1(Xi ∈ B)
.
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![By stationarity, the last quantity does not depend on the particular choice of set B. Let g(x, Φ̃) = 1(x ∈
B)e0δ0. The right hand side of the above equation can be expressed as
1
|B|
E
hZ
R2
g(x, Φ̃ − x) Φ(dx)
i
,
and by Campbell’s formula (2.9 in Volume I) is equal to
λ
|B|
Z
R2
E0
[g(x, Φ̃)] dx = λE0
[e0δ0] ,
which gives the following result
dsuc(r, λ1, T) = λ1pc(r, λ1, T) = λppc(r, λp, T). (16.21)
Similarly, the density of progress can be defined as the mean distance progressed by all transmissions
taking place in some arbitrary subset B of the plane
dprog(r, λ1, T) =
1
|B|
E
X
i
reiδi1(Xi ∈ B)
= rλ1pc(r, λ1, T) , (16.22)
by the same arguments as for (16.21).
The spatial density of throughput is equal to
dthrou(r, λ1) =
1
|B|
E
X
i
ei1(Xi ∈ B) log(1 + SINRi)
= λ1τ(r, λ1) (16.23)
and the density of transport to
dtrans(r, λ1) =
1
|B|
E
X
i
eir1(Xi ∈ B) log(1 + SINRi)
= λ1rτ(r, λ1). (16.24)
In the following sections we focus on the optimization of the spatial performance of an Aloha MANET.
16.3.1 Optimization of the Density of Progress
16.3.1.1 Best MAP Given Some Transmission Distance
We already mentioned that a good tuning of p should find a compromise between the average number of
concurrent transmissions per unit area and the probability that a given authorized transmission is successful.
To find such a compromise, one ought to maximize the density of progress, or equivalently the density of
successful transmissions, dsuc(r, λp, T) = λp pc(r, λp, T), w.r.t. p, for a given r and λ. This can be done
explicitly for the Poisson bipolar network model with Rayleigh fading. For this we first optimize w.r.t. λ
assuming p fixed and then deduce from this the optimal MAP for some fixed λ.
Define
λmax = arg max
0≤λ∞
dsuc(r, λ, T) ,
whenever such a value of λ exists and is unique. The following result follows from Proposition 16.2.2.
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![0
0.005
0.01
0.015
0.02
0.025
0.03
0 0.05 0.1 0.15 0.2
Density
of
successful
transmissions
d
suc
MAP p
Rayleigh
Rician q=0.5
Rician q=0.9
Fig. 16.2 Density of successful transmissions dsuc for Aloha in function of p in the Rayleigh and the Rician (with q = 1/2 and q = .9) fading
cases; λ = r = 1, W = 0, and T = 10dB; we use OPL 3 with A = 1 and β = 4.
Example 16.3.5 (Rayleigh versus Rician fading). Figure 16.2 compares the density of success for
Rayleigh and Rician fading. For this, we use the representations of Propositions 16.2.2 and Proposition
16.2.4, respectively. In the Rayleigh case, F is exponential with mean 1. In the Rician case F = q+(1−q)F0,
with 0 ≤ q ≤ 1, where F0 is exponential with mean 1 and q represents the part of the energy received on the
line-of-sight. The density of success is plotted in function of p. We again observe that higher variances are
beneficial for high densities of transmitters (which is here equivalent to the far field case) and detrimental for
low densities. However here, in each case, there is an optimal MAP, and when properly optimized, Spatial
Aloha does better for lower variances (i.e. for Rician fading with higher q).
16.3.1.2 *General Definition of λmax
In this section, we show that under some mild conditions, λmax is well defined and not degenerate (i.e.
0 λmax ∞) for a general GI
W+M/GI model. Assume T 0. Note that dsuc(r, 0, T) = 0; so under some
natural non-degeneracy assumptions, the maximum is certainly not attained at λ = 0.
Proposition 16.3.6. Consider the GI
W+M/GI bipolar model with p = 1 and general fading with a finite mean.
Assume that l(r) 0 and is such that the generic SN I(λ) =
P
Xj∈e
Φ
Fj/l(|Xj|) admits a density for all
λ 0. Then
(1) If P{ F 0 } 0, then pc(r, x, T) (and so dsuc(r, x, T)) is continuous in x, so that the maxi-
mum of the function x → dsuc(r, x, T) in the interval [0, λ] is attained for some 0 λmax ≤ λ;
(2) If for all a 0, the following modified SN:
I0
(λ) =
X
Xj∈e
Φ
1(|Xj| a)Fj/l(|Xj|)
3The question of the definition of λmax is addressed in § 16.3.1.2.
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![has finite mean for all λ 0, then limx→∞ dsuc(r, x, T) = 0 and consequently, for sufficiently
large λ, this maximum is attained for some λmax λ.
The statement of the last proposition means that for a sufficiently large density of nodes λ, a nontrivial
MAP 0 pmax 1 equal to pmax = λmax/λ optimizes the density of successful transmissions.
Proof. (of Proposition 16.3.6) Recall that pc(r, λ, T) = P{ I(λ) ≤ F/(l(r)T−W) }, where the dependence
of the SN variable I(λ) = I1 = I (note that p = 1) on the intensity of the Poisson p.p. was made explicit.
By the thinning property of the Poisson p.p. (stated in Proposition 1.3.5 in Volume I) we can split the SN
variable into two independent Poisson SN terms I(λ + ) = I(λ) + I(). Moreover, we can do this in such
a way that I(), which is finite by assumption, almost surely converges to 0 when → 0. Consequently,
0 ≤ pc(r, λ, T) − pc(r, λ + , T) = P
n F
l(r)T − W
− I() I(λ) ≤
F
l(r)T − W
o
,
and
lim
→0
pc(r, λ, T) − pc(r, λ + , T)
= P
n
I(λ) =
F
l(r)T − W
o
= 0 ,
where the last equation is due to the fact that I(λ) is independent of F, W and admits a density. Splitting the
Poisson p.p. of intensity λ into two Poisson p.p.s with intensity λ − and respectively, and considering the
associated SN variables I(λ−) and I(), with I(λ) defined as their sum, one can show in a similar manner
that lim→0 pc(r, λ − , T) − pc(r, λ, T) = 0. This concludes the proof of the first part of the proposition.
We now prove the second part. Let G(s) = P{ F ≥ s }. Take 0 and such that E[I0(1)] =
I
0
(1) ∞. By independence we have
dsuc(r, λ, T) = λpc(r, λ, T) ≤ λE
h
P
n
F ≥ I0
(λ)Tl(r)|I0
(λ)
oi
≤ J1 + J2 ,
where
J1 = E
h λ
I0(λ)
1
I0
(λ) ≥ λ(I
0
(1) − )
I0
(λ)G I0
(λ)Tl(r)
i
J2 = λE
h
1
I0
(λ) λ(I
0
(1) − )
i
.
Since E[F] =
R ∞
0 G(s) ds ∞, I0(λ)G I0(λ)Tl(r)
is uniformly bounded in I0(λ) and
I0(λ)G I0(λ)Tl(r)
→ 0 when I0(λ) → ∞. Moreover, one can construct a probability space such that
I0(λ) → ∞ almost surely as λ → ∞. Thus, by Lebesgue’s dominated convergence theorem, we have
limλ→∞ J1 = 0.
For J2 and t 0 we have
J2 ≤ λP0
{ e−tI0(λ)
≥ e−λt(I
0
(1)−)
≤ λE0
e−tI0(λ)+λt(I
0
(1)−)
]
= λ exp
λ
t(I
0
(1) − ) − 2π
∞
Z
a
s 1 − LF (t/l(s))
ds
.
Note that the derivative of t(I
0
(1) − ) − 2π
R ∞
a s 1 − LF (t/l(s))
ds with respect to t at t = 0 is equal to
I
0
(1) − − I
0
(1) 0. Thus, for some small t 0, J2 ≤ λe−λC for some constant C 0. This shows that
limλ→∞ J2 = 0, which concludes the proof.
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![Proposition 16.4.2. Assume the Poisson bipolar network model of Section 16.2.1 with opportunistic Aloha
MAC given by (2’) in Section 16.4.1 for some general distribution of F and θ. Take the same assumptions
as in Proposition 16.2.4 except that the condition on F is replaced by the following one
• E[Fθ] ∞ and Fθ admits a square integrable density.
Then
b
pc(r, λ1, T) =
∞
Z
−∞
LI1 (2iπl(r)Ts) LW (2iπl(r)Ts)
LFθ
(−2iπs) − 1
2iπs
ds . (16.44)
Proof. It follows immediately from Equation (16.42) and Corollary 12.2.2 in Volume I.
In the case of Rayleigh fading and deterministic θ, the last theorem can be used since Fθ is then the
convolution of a deterministic distribution (with mean θ) and an exponential one (with parameter µ); this
satisfies the assumptions of the proposition. In this case, LFθ
(s) = e−sθ µ
µ+s . Obviously it can also be used
when θ is exponentially distributed with intensity ν. In particular, for Rayleigh fading, we have LFθ
(s) =
µ+ν
µ+ν+s
µ
µ+s.
Example 16.4.3. Assume Rayleigh fading. In Figure 16.3 we plot the density of successful transmissions
dsuc in function of the parameter ν for three different scenarios:
(1) Opportunistic Aloha with a deterministic threshold θ with value 1/ν, where dsuc =
λ1b
pc(r, λ1, ν), with λ1 = λe−µ
ν and b
pc(r, λ1, ν) given by Proposition 16.4.2;
(2) Opportunistic Aloha with a random exponential threshold with parameter ν, where dsuc =
λ1b
pc(r, λ1, ν), with λ1 = λν
µ+ν and b
pc(r, λ, ν) given by Proposition 16.4.1;
(3) Plain Aloha where dsuc = 1/(eK(β)r2T2/β) is the optimal density of successful transmissions
as obtained in Proposition 16.3.2 (this is of course a constant in ν).
In the particular case considered in this figure, the density of transmitters covering their target receiver is
approx. 56% larger in the optimal opportunistic scheme with exponential threshold than in plain Aloha and
134% larger in the deterministic case.
It may look surprising that the curves for the random exponential and the deterministic threshold cases
(1 and 2 above) differ so much. One should bear in mind the fact that the two associated MAPs are quite
different: p = ν
µ+ν in the former case and p = e−µ
ν in the latter.
Example 16.4.4 (Rayleigh versus Rician fading). (Example 16.3.5 continued).
Figure 16.4 compares the density of success of opportunistic Aloha for Rayleigh and Rician (q = 1/2)
fading. (The Rician case with q = .9 has thresholds θ larger than .9 and leads to very small densities
of success; it is not displayed). The two curves are based on Proposition 16.4.2. Note first that for the two
considered cases the density of transmitters are quite different: (exp(−θ) in the Rayleigh case and exp(−2θ)
in the Rician case, for θ 1/2), which explain why the shapes of the curves are so different. Here, we see
24
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Stochastic Geometry And Wireless Networks Part Ii Applications Francois Baccelli
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- 6. Stochastic Geometry and Wireless Networks Volume II APPLICATIONS François Baccelli and Bartlomiej Blaszczyszyn INRIA & Ecole Normale Supérieure, 45 rue d'Ulm, Paris.
- 7. This monograph is based on the lectures and tutorials of the authors at Université Paris 6 since 2005, Eu- random (Eindhoven, The Netherlands) in 2005, Performance 05 (Juan les Pins, France), MIRNUGEN (La Pedrera, Uruguay) and Ecole Polytechnique (Palaiseau, France) in 2007.
- 8. To Béatrice and Mira i inria-00403040, version 4 - 4 Dec 2009
- 10. Preface A wireless communication network can be viewed as a collection of nodes, located in some domain, which can in turn be transmitters or receivers (depending on the network considered, nodes may be mobile users, base stations in a cellular network, access points of a WiFi mesh etc.). At a given time, several nodes transmit simultaneously, each toward its own receiver. Each transmitter–receiver pair requires its own wireless link. The signal received from the link transmitter may be jammed by the signals received from the other transmitters. Even in the simplest model where the signal power radiated from a point decays in an isotropic way with Euclidean distance, the geometry of the locations of the nodes plays a key role since it determines the signal to interference and noise ratio (SINR) at each receiver and hence the possibility of establishing simultaneously this collection of links at a given bit rate. The interference seen by a receiver is the sum of the signal powers received from all transmitters, except its own transmitter. Stochastic geometry provides a natural way of defining and computing macroscopic properties of such networks, by averaging over all potential geometrical patterns for the nodes, in the same way as queuing theory provides response times or congestion, averaged over all potential arrival patterns within a given parametric class. Modeling wireless communication networks in terms of stochastic geometry seems particularly relevant for large scale networks. In the simplest case, it consists in treating such a network as a snapshot of a stationary random model in the whole Euclidean plane or space and analyzing it in a probabilistic way. In particular the locations of the network elements are seen as the realizations of some point processes. When the underlying random model is ergodic, the probabilistic analysis also provides a way of estimating spatial averages which often capture the key dependencies of the network performance characteristics (connectivity, stability, capacity, etc.) as functions of a relatively small number of parameters. Typically, these are the densities of the underlying point processes and the parameters of the protocols involved. By spatial average, we mean an empirical average made over a large collection of ’locations’ in the domain considered; depending on the cases, these locations will simply be certain points of the domain, or nodes located in the domain, or even nodes on a certain route defined on this domain. These various kinds of iii inria-00403040, version 4 - 4 Dec 2009
- 11. spatial averages are defined in precise terms in the monograph. This is a very natural approach e.g. for ad hoc networks, or more generally to describe user positions, when these are best described by random processes. But it can also be applied to represent both irregular and regular network architectures as observed in cellular wireless networks. In all these cases, such a space average is performed on a large collection of nodes of the network executing some common protocol and considered at some common time when one takes a snapshot of the network. Simple examples of such averages are the fraction of nodes which transmit, the fraction of space which is covered or connected, the fraction of nodes which transmit their packet successfully, and the average geographic progress obtained by a node forwarding a packet towards some destination. This is rather new to classical performance evaluation, compared to time averages. Stochastic geometry, which we use as a tool for the evaluation of such spatial averages, is a rich branch of applied probability particularly adapted to the study of random phenomena on the plane or in higher dimension. It is intrinsically related to the theory of point processes. Initially its development was stimulated by applications to biology, astronomy and material sciences. Nowadays, it is also used in image analysis and in the context of communication networks. In this latter case, its role is similar to that played by the theory of point processes on the real line in classical queuing theory. The use of stochastic geometry for modeling communication networks is relatively new. The first papers appeared in the engineering literature shortly before 2000. One can consider Gilbert’s paper of 1961 (Gilbert 1961) both as the first paper on continuum and Boolean percolation and as the first paper on the analysis of the connectivity of large wireless networks by means of stochastic geometry. Similar observations can be made on (Gilbert 1962) concerning Poisson–Voronoi tessellations. The number of papers using some form of stochastic geometry is increasing fast. One of the most important observed trends is to take better account in these models of specific mechanisms of wireless communications. Time averages have been classical objects of performance evaluation since the work of Erlang (1917). Typical examples include the random delay to transmit a packet from a given node, the number of time steps required for a packet to be transported from source to destination on some multihop route, the frequency with which a transmission is not granted access due to some capacity limitations, etc. A classical reference on the matter is (Kleinrock 1975). These time averages will be studied here either on their own or in conjunction with space averages. The combination of the two types of averages unveils interesting new phenomena and leads to challenging mathematical questions. As we shall see, the order in which the time and the space averages are performed matters and each order has a different physical meaning. This monograph surveys recent results of this approach and is structured in two volumes. Volume I focuses on the theory of spatial averages and contains three parts. Part I in Volume I provides a compact survey on classical stochastic geometry models. Part II in Volume I focuses on SINR stochastic geometry. Part III in Volume I is an appendix which contains mathematical tools used throughout the monograph. Volume II bears on more practical wireless network modeling and performance analysis. It is in this volume that the interplay between wireless communications and stochastic geometry is deepest and that the time–space framework alluded to above is the most important. The aim is to show how stochastic geometry can be used in a more or less systematic way to analyze the phenomena that arise in this context. Part IV in Volume II is focused on medium access control (MAC). We study MAC protocols used in ad hoc networks and in cellular networks. Part V in Volume II discusses the use of stochastic geometry for the iv inria-00403040, version 4 - 4 Dec 2009
- 12. quantitative analysis of routing algorithms in MANETs. Part VI in Volume II gives a concise summary of wireless communication principles and of the network architectures considered in the monograph. This part is self-contained and readers not familiar with wireless networking might either read it before reading the monograph itself, or refer to it when needed. Here are some comments on what the reader will obtain from studying the material contained in this monograph and on possible ways of reading it. For readers with a background in applied probability, this monograph provides direct access to an emerg- ing and fast growing branch of spatial stochastic modeling (see e.g. the proceedings of conferences such as IEEE Infocom, ACM Sigmetrics, ACM Mobicom, etc. or the special issue (Haenggi, Andrews, Baccelli, Dousse, and Franceschetti 2009)). By mastering the basic principles of wireless links and of the organi- zation of communications in a wireless network, as summarized in Volume II and already alluded to in Volume I, these readers will be granted access to a rich field of new questions with high practical interest. SINR stochastic geometry opens new and interesting mathematical questions. The two categories of objects studied in Volume II, namely medium access and routing protocols, have a large number of variants and of implications. Each of these could give birth to a new stochastic model to be understood and analyzed. Even for classical models of stochastic geometry, the new questions stemming from wireless networking often provide an original viewpoint. A typical example is that of route averages associated with a Poisson point process as discussed in Part V in Volume II. Reader already knowledgeable in basic stochastic geometry might skip Part I in Volume I and follow the path: Part II in Volume I ⇒ Part IV in Volume II ⇒ Part V in Volume II, using Part VI in Volume II for understanding the physical meaning of the examples pertaining to wireless networks. For readers whose main interest in wireless network design, the monograph aims to offer a new and comprehensive methodology for the performance evaluation of large scale wireless networks. This method- ology consists in the computation of both time and space averages within a unified setting. This inherently addresses the scalability issue in that it poses the problems in an infinite domain/population case from the very beginning. We show that this methodology has the potential to provide both qualitative and quantitative results as below: • Some of the most important qualitative results pertaining to these infinite population models are in terms of phase transitions. A typical example bears on the conditions under which the network is spatially connected. Another type of phase transition bears on the conditions under which the network delivers packets in a finite mean time for a given medium access and a given routing protocol. As we shall see, these phase transitions allow one to understand how to tune the protocol parameters to ensure that the network is in the desirable ”phase” (i.e. well connected and with small mean delays). Other qualitative results are in terms of scaling laws: for instance, how do the overhead or the end-to-end delay on a route scale with the distance between the source and the destination, or with the density of nodes? • Quantitative results are often in terms of closed form expressions for both time and space aver- ages, and this for each variant of the involved protocols. The reader will hence be in a position v inria-00403040, version 4 - 4 Dec 2009
- 13. to discuss and compare various protocols and more generally various wireless network organiza- tions. Here are typical questions addressed and answered in Volume II: is it better to improve on Aloha by using a collision avoidance scheme of the CSMA type or by using a channel-aware ex- tension of Aloha? Is Rayleigh fading beneficial or detrimental when using a given MAC scheme? How does geographic routing compare to shortest path routing in a mobile ad hoc network? Is it better to separate the medium access and the routing decisions or to perform some cross layer joint optimization? The reader with a wireless communication background could either read the monograph from beginning to end, or start with Volume II i.e. follow the path Part IV in Volume II ⇒ Part V in Volume II ⇒ Part II in Volume I and use Volume I when needed to find the mathematical results which are needed to progress through Volume II. We conclude with some comments on what the reader will not find in this monograph: • We do not discuss statistical questions and give no measurement based validation of certain stochastic assumptions used in the monograph: e.g. when are Poisson-based models justified? When should one rather use point processes with some repulsion or attraction? When is the sta- tionarity/ergodicity assumption valid? Our only aim is to show what can be done with stochastic geometry when assumptions of this kind can be made. • We will not go beyond SINR models either. It is well known that considering interference as noise is not the only possible option in a wireless network. Other options (collaborative schemes, suc- cessive cancellation techniques) can offer better rates, though at the expense of more algorithmic overhead and the exchange of more information between nodes. We believe that the methodology discussed in this monograph has the potential of analyzing such techniques but we decided not to do this here. Here are some final technical remarks. Some sections, marked with a * sign, can be skipped at the first reading as their results are not used in what follows; The index, which is common to the two volumes, is designed to be the main tool to navigate within and between the two volumes. Acknowledgments The authors would like to express their gratitude to Dietrich Stoyan, who first suggested them to write a monograph on this topic, as well as to Daryl Daley and Martin Haenggi for their very valuable proof-reading of the manuscript. They would also like to thank the anonymous reviewer of NOW for his suggestions, particularly so concerning the two volume format, as well as Paola Bermolen, Pierre Brémaud, Srikant Iyer, Mohamed Karray, Omid Mirsadeghi, Paul Muhlethaler, Barbara Staehle and Patrick Thiran for their useful comments on the manuscript. vi inria-00403040, version 4 - 4 Dec 2009
- 14. Preface to Volume II The two first parts of volume II (Part IV and Part V) are structured in terms of the key ingredients of wireless communications, namely medium access and routing. The general aim of this volume is to show how stochastic geometry can be used in a more or less systematic way to analyze the key phenomena that arise in this context. We limit ourselves to simple (yet not simplistic) models and to basic protocols. This volume is nevertheless expected to convince the reader that much more can be done for improving the realism of the models, for continuing the analysis and for extending the scope of the methodology. Part IV is focused on medium access control (MAC). We study MAC protocols used both in mobile ad hoc networks (MANETs) and in cellular networks. We analyze spatial Aloha schemes in terms of Poisson shot-noise processes in Chapters 16 and 17 and carrier sense multiple access (CSMA) schemes in terms of Matérn point processes in Chapter 18. The analytical results are then used to perform various optimizations on these schemes. For instance, we determine the tuning of the protocol parameters which maximizes the number of successful transmissions or the throughput per unit of space. We also determine the protocol parameters for which end-to-end delays have a finite mean, etc. Chapter 19 is focused on the Code Division Multiple Access (CDMA) schemes with power control which are used in cellular networks. The terminal nodes associated with a given concentration node (base station, access point) are those located in its Voronoi cell w.r.t. the point process of concentration nodes. For analyzing these systems, we use both shot noise processes and tessellations. When the terminal nodes require a fixed bit rate, and power is controlled so as to maximize the number of terminal nodes that can be served by such a cellular network, powers become functionals of the underlying point processes. We study admission control and capacity within this context. Part V discusses the use of stochastic geometry for the qualitative and quantitative analysis of routing algorithms in a MANET where the nodes are some realization of a Poisson point process (p.p.) of the plane. In the point-to-point routing case, the main object of interest is the path from some source to some destination node. In the point-to-multipoint case, this is the tree rooted in the source node and spanning a set of destination nodes. The motivations are multihop diffusion in MANETs. We also analyze the multipoint-to-point case, which is used for instance for concentration in wireless sensor communication vii inria-00403040, version 4 - 4 Dec 2009
- 15. networks where information has to be gathered at some central node. These random geometric objects are made of a set of wireless links, which have to be either simultaneously of successively feasible. Chapter 20 is focused on optimal routing, like e.g. shortest path and minimal weight routing. The main tool is subadditive ergodic theory. In Chapter 21, we analyze various types of suboptimal (greedy) geographic routing schemes. We show how to use stochastic geometry to analyze local functionals of the random paths/tree such as the distribution of the length of its edges or the mean degree of its nodes. Chapter 22 bears on time-space routing. This class of routing algorithms leverages the interaction between MAC and routing and belongs to the so called cross-layer framework. More precisely, these algorithms take advantage of the time and space diversity of fading variables and MAC decisions to route packets from source to destination. Typical qualitative results bear on the ’convergence’ of these routing algorithms or on the fact that the velocity of a packet on a route is positive or zero. Typical quantitative results are in terms of the comparison of the mean time it takes to transport a packet from some source node to some destination node. Part VI is an appendix which contains a concise summary of wireless communication principles and of the network architectures considered in the monograph. Chapter 23 is focused on propagation issues and on statistical channel models for fading such as Rayleigh or Rician fading. Chapter 24 bears on detection with a special focus on the fundamental limitations of wireless channels. As for architecture, we describe both MANETs and cellular networks in Chapter 25. MANETs are “flat” networks, with a single type of nodes which are at the same time transmitters, receivers and relays. Examples of MAC protocols used within this framework are described as well as multihop routing principles. Cellular networks have two types of network elements: base stations and users. Within this context, we discuss power control and its feasibility as well as admission control. We also consider other classes of hetero- geneous networks like WiFi mesh networks, sensor networks or combinations of WiFi and cellular networks. Let us conclude with a few general comments on the wireless channels and the networks to be considered throughout the volume. 1 Two basic communication models are considered: • A digital communication model, where the throughput on a link (measured in bits per seconds) is determined by the SINR at the receiver through a Shannon-like formula; • A packet model, where the SINR at the receiver determines the probability of reception (also called probability of capture) of the packet and where the throughput on a link is measured in packets per time slot. In most models, time is slotted and the time slot is assumed to be such that fading is constant over a time slot (see Chapter 23 for more on the physical meaning of this assumption). There are hence at least three time scales: • The time scale of symbol transmissions. In this volume, this time scale is considered small com- pared to the time slot, so that many symbols are sent during one slot. At this time scale, the additive noise is typically assumed to be a Gaussian white noise and spreading techniques can be invoked to justify the representation of the interference on each channel as a Gaussian addi- 1For those not familiar with wireless networks, a full understanding of these comments might require a preliminary study of Part VI viii inria-00403040, version 4 - 4 Dec 2009
- 16. tive white noise (see § 24.3.3). Shannon’s formula can then in turn be invoked to determine the bit-rate of each channel over a given time slot in terms of the ratio of the mean signal power to the mean interference-and-noise power seen on the channel; the latter mean is the sum of the variance of noise and of the variance of the Gaussian representation of interference; the bit rate is an ergodic average over the many symbols sent in one slot. • The time scale of slots. At this time scale, only the mean interference and noise powers for each channel and each time slot are retained from the symbol transmission time scale. These quantities change from a time slot to the next due to the fact that MAC decisions and fading may change. For example, with Aloha, the MAC decisions are resampled at each time slot; as for fading, we consider a fast fading scenario, 2 where the fading between a transmitter and a receiver changes (e.g. is resampled) from a time slot to the next (for instance do the motion of reflectors – see § 23) and a slow fading scenario, where it remains unchanged over time slots. At this time scale, the interference powers are hence again random processes, fully determined by the fading scenario and the MAC. As we shall see, their laws (which are not Gaussian anymore) can be determined using the Shot-Noise theory of Chapter 2 in Volume I. • The time scale of mobility. In this monograph, this time scale is considered large compared to time slots. In particular in the part on routing, we primarily focus on scenarios where all nodes are static and where routes are established on this static network. The rationale is that the time scale of packet transmission on a route is smaller than that of node mobility. Stated differently, we do not consider here the class of delay tolerant networks which leverage node mobility for the transport of packets. 2Notice that this definition of fast fading differs from the definition used in many papers of literature, where fast fading often means that the channel conditions fluctuate much over a given time slot. ix inria-00403040, version 4 - 4 Dec 2009
- 18. Contents of Volume II Preface iii Preface to Volume II vii Contents of Volume II xi Part IV Medium Access Control 1 16 Spatial Aloha: the Bipole Model 3 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 16.2 Spatial Aloha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 16.3 Spatial Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 16.4 Opportunistic Aloha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 16.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 17 Receiver Selection in Spatial Aloha 29 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 17.2 Nearest Receiver Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 17.3 Multicast Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 17.4 Opportunistic Receivers – MAC and Routing Cross-Layer Optimization . . . . . . . . . . . 39 17.5 Local Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 17.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 18 Carrier Sense Multiple Access 69 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 18.2 Matérn-like Point Process for Networks using Carrier Sense Multiple Access . . . . . . . . 70 18.3 Probability of Medium Access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 xi inria-00403040, version 4 - 4 Dec 2009
- 19. 18.4 Probability of Joint Medium Access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 18.5 Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 18.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 19 Code Division Multiple Access in Cellular Networks 79 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 19.2 Power Control Algebra in Cellular Networks . . . . . . . . . . . . . . . . . . . . . . . . . . 80 19.3 Spatial Stochastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 19.4 Maximal Load Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 19.5 A Time–Space Loss Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 19.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Bibliographical Notes on Part IV 99 Part V Multihop Routing in Mobile ad Hoc Networks 101 20 Optimal Routing 105 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 20.2 Optimal Multihop Routing on a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 20.3 Asymptotic Properties of Minimal Weight Routing . . . . . . . . . . . . . . . . . . . . . . 107 20.4 Largest Bottleneck Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 20.5 Optimal Multicast Routing on a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 20.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 21 Greedy Routing 115 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 21.2 Examples of Geographic Routing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 116 21.3 Next-in-Strip Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 21.4 Smallest Hop Routing — Spatial Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 21.5 Smallest Hop Multicast Routing — Spatial Averages . . . . . . . . . . . . . . . . . . . . . 123 21.6 Smallest Hop Directional Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 21.7 Space and Route Averages of Smallest Hop Routing . . . . . . . . . . . . . . . . . . . . . . 127 21.8 Radial Spanning Tree in a Voronoi Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 21.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 22 Time-Space Routing 133 22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 22.2 Stochastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 22.3 The Time–Space Signal-to-Interference Ratio Graph and its Paths . . . . . . . . . . . . . . 135 22.4 Routing and Time–Space Point Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 22.5 Minimal End-to-End Delay Time–Space Paths . . . . . . . . . . . . . . . . . . . . . . . . . 143 22.6 Opportunistic Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 22.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 xii inria-00403040, version 4 - 4 Dec 2009
- 20. Bibliographical Notes on Part V 163 Part VI Appendix: Wireless Protocols and Architectures 165 23 Radio Wave Propagation 167 23.1 Mean Power Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 23.2 Random Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 23.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 24 Signal Detection 179 24.1 Complex Gaussian Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 24.2 Discrete Baseband Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 24.3 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 24.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 25 Wireless Network Architectures and Protocols 189 25.1 Medium Access Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 25.2 Power Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 25.3 Examples of Network Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Bibliographical Notes on Part VI 197 Bibliography 199 Table of Notation 203 Index 205 xiii inria-00403040, version 4 - 4 Dec 2009
- 21. Part IV Medium Access Control 1 inria-00403040, version 4 - 4 Dec 2009
- 22. In this part, we analyze various kinds of medium access control (MAC) protocols using stochastic geom- etry tools. The reader not familiar with MAC should refer to §25.1 in the appendix in case what is described in the chapters is not self-sufficient. We begin with Aloha, which is analyzed in detail in Chapters 16 and 17 (Aloha is also central in Chap- ter 22). We then analyze CSMA in Chapter 18, both in the context of MANETs. We then study CDMA in cellular networks in Chapter 19. In all three cases, we develop a whole-plane snapshot analysis which yields estimates of various instantaneous spatial or time-space performance metrics. This whole-plane analysis is meant to address the scalability of protocols since it focuses on results which are pertinent for very large networks. For MANETs using Aloha and CSMA, we analyze: • The spatial density of nodes authorized to transmit by the MAC; • The probability of success of a typical transmission, which leads to formulas for the density of successful transmissions whenever a target SINR is prescribed; • The distribution of the throughput obtained by an authorized node in case of elastic traffic, namely when no target SINR is given, which leads to estimates for the density of throughput in the network, where the throughput is defined in terms of a Shannon-like formula. In addition to this digital communication view point, we also discuss the packet transmission model, where the throughput is defined in terms of the number of time slots required to successfully transmit a packet. For the CDMA case, we incorporate the key concept of power control (see § 25.2 in the appendix for the algebraic formulation of the power control problem). In this case, the spatial performance metrics are: • For the case where a target SINR is prescribed, the spatial intensity of cells where some admission control has to be enforced in order to make the global power control problem feasible and the resulting density of users accepted in the access network; • For the case where no target SINR is prescribed, the density of throughput in the network. A key paradigm throughout this part is that of the ‘social optimization’ of the protocol, which consists in determining the tuning of the protocol parameters which maximizes the density of successful transmissions or the density of throughput or the spatial reuse (to be defined) in such a network. Let us stress that the analysis of Aloha is by far the most comprehensive and that the aim of the CSMA and the CDMA chapters is primarily to introduce features (spatial contention for the former and power control for the latter) which are not present in Aloha. As we shall see, these new features lead to technical difficulties and more research will be required to extend all types of results available for Aloha to these more complex (and more realistic) MAC protocols. 2 inria-00403040, version 4 - 4 Dec 2009
- 23. 16 Spatial Aloha: the Bipole Model 16.1 Introduction In this chapter we consider a simplified MANET model called the bipolar model, where we do not yet address routing issues but do assume that each transmitter has its receiver at some fixed or random distance. We study a slotted version of Aloha. As explained in Section 25.1.2, under the Aloha MAC protocol, at each time slot, each potential transmitter independently tosses a coin with some bias p which will be referred to as the medium access probability (MAP); it accesses the medium if the outcome is heads and it delays its transmission otherwise. It is important to tune the value of the MAP p so as to realize a compromise between the two contradict- ing wishes to have a large average number of concurrent transmissions per unit area and a high probability that authorized transmissions be successful: large values of p allow more concurrent transmissions but (sta- tistically) smaller exclusion zones, making these transmissions more vulnerable; smaller values of p give fewer transmissions with higher probability of success. Another important tradeoff concerns the typical one-hop distance of transmissions. A small distance makes the transmissions more secure but involves more relaying nodes to communicate packets between origin and destination. On the other hand, a larger one-hop distance reduces the number of hops but might increase the number of failed transmissions and hence that of retransmissions at each hop. We define and study the following performance metrics: the probability of success (high enough SINR) for a typical transmission and the mean throughput (bit-rate) for a typical transmission at this distance. The mean number of successful transmissions and the mean throughput per unit area, the mean distance of progress per unit area, the transport density, etc. are studied in § 16.3, together with the notion of spatial reuse which is quite useful to compare scenarios and policies. Let us stress that the definitions of § 16.3 extend to other MAC than Aloha and will be used throughout the present volume. In § 16.4, this simplified setting is also used to study variants of Aloha such as opportunistic Aloha, which leverages the channels fluctuations due to fading. 3 inria-00403040, version 4 - 4 Dec 2009
- 24. 16.2 Spatial Aloha 16.2.1 The Poisson Bipolar MANET Model with Independent Fading and Aloha MAC Below, we consider a Poisson bipolar network model in which each point of the Poisson pattern represents a node of the MANET (a potential transmitter) and has an infinite backlog of packets to transmit to its associated receiver; the latter is not part of the Poisson pattern of points and is located at distance r. 1 This model can be used to represent a snapshot of a multi-hop MANET: the transmitters are some relay nodes and are not necessarily the sources of the transmitted packets; similarly, the receivers need not be the final destinations. More precisely, we assume that the MANET snapshot can be represented by an independently marked (i.m.) Poisson p.p. (cf. Sections 1.1.1 and 2.1.1 in Volume I), where the Poisson p.p. is homogeneous on the Euclidean plane, with intensity λ, and where the multidimensional mark of a point carries information about the MAC status of the point (allowed to transmit or delayed; cf. Section 25.1) at the current time slot and about the fading conditions of the channels to all receivers (cf. Section 23.2 and also Section 2.3 in Volume I). This marked Poisson p.p. is denoted by e Φ = {(Xi, ei, yi, Fi)}, where (1) Φ = {Xi} denotes the locations of the points (the potential transmitters); Φ is always assumed Poisson with positive and finite intensity λ; (2) {ei} is the medium access indicator of node i; (ei = 1 if node i is allowed to transmit in the considered time slot and 0 otherwise). The random variables ei are hence i.i.d. and independent of everything else, with P(ei = 1) = p (p is the MAP). (3) {yi} denotes the location of the receiver for node Xi (we assume here that no two transmitters have the same receiver). We assume that the random vectors {Xi − yi} are i.i.d with |Xi − yi| = r; i.e. each receiver is at distance r from its transmitter (see Figure 16.1). There is no difficulty extending what is described below to the case where these distances are independent and identically distributed random variables, independent of everything else. (4) {Fi = (Fj i : j)} where Fj i denotes the virtual power emitted by node i (provided ei = 1) towards receiver yj. By virtual power Fj i , we understand the product of the effective power of transmitter i and of the random fading from this node to receiver yj (cf. Remark 2.3.1 in Volume I). The random vectors {Fi} are assumed to be i.i.d. and the components (Fj i , j) are assumed to be identically distributed (distributed as a generic random variable (r.v.) denoted by F) with mean 1/µ assumed finite. In the case of constant effective transmission power 1/µ and Rayleigh fading, F is exponential with mean 1/µ (see Section 23.2.4). In this case, it is reasonable to assume that the components of (Fj i : j) are independent, which is the default option in what follows. This is justified if the distance between two receivers is larger than the coherence distance of the wireless channel (cf. Section 23.3), which is a natural assumption here. Below, we also consider non-exponential cases which allow one to analyze other types of fading scenarios such as e.g. Rician or Nakagami (see Section 23.2.4) or simply the case without fading (when F ≡ 1/µ is deterministic). In addition, we consider a non-negative random variable W independent of e Φ modeling the power of the thermal noise. A natural extension consists in considering a random field rather than a random variable. Since we assume that Aloha is used, the set of nodes that transmit in the reference time slot Φ1 = 1The fact that all receivers are at the same distance from their transmitter is a simplification that will be relaxed in Chapter 16. 4 inria-00403040, version 4 - 4 Dec 2009
- 25. 1 e= Emitter e=0 Silent node r Receiver Fig. 16.1 A snapshot of bipolar MANET with Aloha MAC. {Xi : ei = 1} is an independent thinning of Φ; By the corresponding property of the Poisson p.p. (cf. Proposition 1.3.5 in Volume I), Φ1 is a Poisson p.p. with intensity λ1 = λp. Select some omni-directional path-loss (OPL) model l(·) (see Section 23.1.2). An important special case consists in taking l(u) = (Au)β for A > 0 and β > 2, (16.1) which we call OPL 3. Note that 1/l(u) has a pole at u = 0, and thus OPL 3 is not appropriate for small distances (and hence in particular for u small compared to the mean distance to the nearest neighbor in the Poisson p.p., namely 1/2 √ λ — see Example 1.4.7 in Volume I). Another consequence of this path-loss model is that the total power received at a given location from an infinite Poisson pattern of transmitters has an infinite mean (where the averaging is taken over all configurations of transmitters); cf. Remark 2.3.5 in Volume I. In spite of these drawbacks, the OPL 3 path-loss model (16.1), will be used as the default model in what follows, because it is precise enough for large enough values of u, it simplifies many calculations and it reveals important scaling laws (see Section 16.2.4). Other possible choices of path-loss function, OPL 1, OPL 2, avoiding the pole at at u = 0 are considered in Example 23.1.3. Note that the power received from transmitter (node) j by the receiver of node i is then equal to Fj i /l(|Xj − yi|), where | · | denotes the Euclidean distance on the plane. 16.2.2 Coverage (Non-Outage) Probability for a Typical Node In what follows we present the basic analysis of the performance of the bipolar MANET model. This analysis uses the shot-noise (SN) interference model of Section 2.3 in Volume I and in particular the random cross- fading model model of Example 2.3.9 in Volume I. We assume that a connection between a transmitter and its receiver is successful (or that the formet covers the latter) when the signal to interference and noise ratio (SINR) at the receiver is larger than some threshold T: 5 inria-00403040, version 4 - 4 Dec 2009
- 26. Definition 16.2.1. Transmitter {Xi} covers its receiver yi in the reference time slot if SINRi = Fi i /l(|Xi − yi|) W + I1 i ≥ T , (16.2) where the interference I1 i is the SN associated with e Φ1, namely, I1 i = P Xj∈e Φ1, j6=i Fi j /l(|Xj − yi|) and where T is some SINR threshold. The last condition might be required in practice for xi to be successfully received by yi due to the use of a particular coding scheme associated with a given bit-rate (cf. Section 24.3.4). It is also called the non-outage or the capture condition depending on the framework. Remark: Later, in Section 16.2.3, we shall also consider adaptive coding schemes in which the appropriate choice of coding scheme is selected for each observed SINR level, which allows one to obtain a bit-rate close to that given by Shannon’s law for all such SINR. In what follows we focus on the probability that this property holds true for the typical node of the MANET, given it is a transmitter. This notion can be formalized using Palm theory for stationary marked point processes (cf. Section 2.1.2 in Volume I). Denote by δi the indicator that (16.2) holds, namely, that location yi is covered by transmitter Xi. We can consider δi as a new mark of Xi. The marked point process e Φ enriched with these marks is still stationary (cf. Definition 2.1.4 in Volume I). However, in contrast to the original marks ei, yi, Fi, given the points of Φ, the random variables {δi} are neither independent nor identically distributed. Indeed, the points of Φ lying in dense clusters have a smaller probability of coverage than more isolated points due to interference; in addition, because of the shot noise variables I1 i , the random variables {δi} are dependent. By probability of coverage of a typical node given it is a transmitter, we understand P0 { δ0 = 1 | e0 = 1} = E0 [δ0 | e0 = 1], where P0 is the Palm probability associated to the (marked) stationary point process e Φ and where δ0 is the mark of the point X0 = 0 a.s. located at the origin 0 under P0. This Palm probability P0 is derived from the original (stationary) probability P by the following relation (cf. Definition 2.1.5 in Volume I) P0 { δ0 = 1 | e0 = 1} = 1 λ1|B| E X i δi1(Xi ∈ B) , where B is an arbitrary subset of the plane and |B| is its surface. Thus, knowing that λ1|B| is the expected number of transmitters in B, the typical node coverage probability is the mean number of transmitters which cover their receivers in any given window B in which we observe our MANET. Note that this mean is based on a double averaging: a mathematical expectation – over all possible realizations of the MANET and, for each realization, a spatial averaging – over all nodes in B. If the underlying point process is ergodic (as it is the case for our i.m. Poisson p.p. Φ̃) the typical node coverage probability can also be interpreted as a spatial average of the number of transmitters which cover their receiver in almost every given realization of the MANET and large B (tending to the whole plane; cf. Proposition 1.6.10 in Volume I). For a stationary i.m. Poisson p.p., the probability P0 can easily be constructed using Slivnyak’s theorem (cf. Theorem 1.4.5 in Volume I): under P0, the nodes of the Poisson MANET and their marks follow the 6 inria-00403040, version 4 - 4 Dec 2009
- 27. distribution e Φ ∪ {(X0 = 0, e0, y0, F0)}, where e Φ is the original stationary i.m. Poisson p.p. (i.e. that seen under the original probability P) and (e0, y0, F0) is a new copy of the mark independent of everything else and distributed like all other i.i.d. marks (ei, yi, Fi) of e Φ under P. (cf. Remark 2.1.7 in Volume I). Note that, under P0, the node at the origin (the typical node), is not necessarily a transmitter; e0 is equal to 1 or 0 with probability p and 1 − p respectively. Denote by pc(r, λ1, T) = E0[δ0 | e0 = 1] the probability of coverage of the typical MANET node given it is a transmitter. It follows from the above construction that this probability only depends on the density of effective transmitters λ1 = λp, on the distance r and on the SINR threshold T; it can be expressed using three independent generic random variables F, I1, W by the following formula: pc(r, λ1, T) = P0 { F0 0 l(r)T(W + I1 0 ) | e0 = 1 } = P{ F ≥ Tl(r)(I1 + W) } . (16.3) Note also that this probability is equal to the one-point coverage probability p0(y0) in the GI W+M/GI SINR cell model of Section 5.3.1 in Volume I associated with the Poisson p.p. intensity λ1. (see the meaning of this Kendall-like notation in Section 5.3 in Volume I). For this reason, our Aloha MANET model is of the GI W+M/GI type, where the GI in the numerator indicates a general distribution for the virtual power of the signal F and where the M/GI in the denominator indicates that the SN interference is generated by a Poisson pattern of interferers (M), with a general distribution (G) for their virtual powers. Special cases of distributions marks are deterministic (D) and exponential (M). We recall that M/· denotes a SN model with a Poisson point process. In what follows, we often use the following explicit formula for the Laplace transform of the generic shot-noise I1 = P Xj∈e Φ1 Fj/l(|Xj|), which is valid in the Poisson p.p. case whenever the random variables Fj are independent copies of the generic fading variable F (cf. Corollary 2.3.8 in Volume I): LI1 (s) = E[e−I1s ] = exp −λ12π ∞ Z 0 t 1 − LF (s/l(t)) dt , (16.4) where LF is the Laplace transform of F. This can be derived from the formula for the Laplace functional of the Poisson p.p. (see Propositions 1.2.2 and 2.2.4 in Volume I). 16.2.2.1 Rayleigh Case The next result bears on the Rayleigh fading case (F exponential with mean 1/µ). Using the independence assumptions, it is easy to see that the right-hand side of (16.3) can be rewritten as pc(r, λ) = pc(r, λ, T) = E h e−µ(Tl(r)(I1+W) i = LI1 (µTl(r))LW (µTl(r)) , (16.5) where LW is the Laplace transform of W and where LI1 can be expressed as follows from (16.4) when using the fact that F is exponential: LI1 (s) = exp −2πλ1 ∞ Z 0 t 1 + µl(t)/s dt . (16.6) Using this observation, one immediately obtains (cf. also Proposition 5.3.3 and Example 5.3.4 in Volume I): 7 inria-00403040, version 4 - 4 Dec 2009
- 28. Proposition 16.2.2. For the M W+M/M bipolar model, pc(r, λ1, T) = LW (µTl(r)) exp − 2πλ1 ∞ Z 0 u 1 + l(u)/(Tl(r)) du . (16.7) In particular if W ≡ 0 and that the path-loss model (16.1) is used then pc(r, λ1, T) = exp(−λ1r2 T2/β K(β)) , (16.8) where K(β) = 2πΓ(2/β)Γ(1 − 2/β) β = 2π2 β sin(2π/β) . (16.9) Example 16.2.3. The above result can be used in the following context: assume one wants to operate a MANET in a regime where each transmitter is guaranteed a SINR at least T with a probability larger than 1 − ε, where ε is a predefined quality of service, or equivalently, where the probability of outage is less then ε. Then, if the transmitter-receiver distance is r, the MAP p should be such that pc(r, λp, T) = 1 − ε. In particular, assuming the path-loss setting (16.1), one should take p = min 1, − ln(1 − ε) λr2T2/βK(β) ≈ min 1, ε λr2T2/βK(β) . (16.10) For example, for T = 10dB 2 and OPL 3 model with β = 4, r = 1, one should take p ≈ min (1, 0.064 ε/λ) . 16.2.2.2 General Fading In what follows, we consider a GI W+M/GI bipolar Aloha MANET model. In this case one can get integral representations for the probability of coverage on the basis of the results of Section 5.3.1 in Volume I. Proposition 16.2.4. Consider a GI W+M/GI bipolar model with general fading variables F such that • F has a finite first moment and admits a square integrable density; • Either I1 or W admit a density which is square integrable. Then the probability of a successful transmission is equal to pc(r, λ1, T) = ∞ Z −∞ LI1 (2iπl(r)Ts) LW (2iπl(r)Ts) LF (−2iπs) − 1 2iπs ds . (16.11) 2A positive real number x is 10 log10(x) dB. 8 inria-00403040, version 4 - 4 Dec 2009
- 29. Remark: Sufficient conditions for I1 to admit a density are given in Proposition 2.2.6 in Volume I. Roughly speaking these conditions require that F be non-null and that the path-loss function l be not constant in any interval. This is satisfied e.g. for the OPL 3 and OPL 2 model, but not for OPL 1 – see Example 23.1.3. Concerning the square integrability of the density, which is equivalent to the integrability of |LI1 (is)|2 (see (Feller 1971, p. 510) and also (2.20 in Volume I)), using (16.4), one can easily check that it is satisfied for the OPL 3 model provided P{F 0} 0. Moreover, under the same conditions |LI1 (is)| is integrable (and so is |LI1 (is)|/|s| for large |s|). Proof. (of Proposition 16.2.4) By the independence of I1 and W in (16.3), the second assumption of Propo- sition 16.2.4 implies that I1 + W admits a density g(·) that is square integrable. The result then follows from pc(r, λ1, T) = P{ (I1 + W)Tl(r) F } , by the Plancherel-Parseval theorem; see e.g. (Brémaud 2002, Th. C3.3, p.157)) and for more details Corol- lary 12.2.2 in Volume I. 16.2.3 Shannon Throughput of a Typical Node In Section 16.2.2 we assumed that a channel could be sustained if the SINR was above some fixed threshold T, which corresponds to the case where some minimum bit rate is required (like in e.g. voice). In this section we consider the situation where there is no minimal requirement on the bit rate T and where the latter depends on the SINR through some Shannon like formula. This is possible with adaptive coding, where a coding with a high bit-rate is used if the SINR is high, whereas a coding with a low bit-rate is used in case of lower SINR. We adopt the following definition. Definition 16.2.5. We define the (Shannon) throughput (bit-rate) of the channel from transmitter Xi to its receiver yi to be Ti = log(1 + SINRi) , (16.12) where SINRi is as in Definition 16.2.1. One can then ask about the throughput of a typical transmitter (or equivalently about the spatial average of the rate obtained by the transmitters), namely τ = τ(r, λ1) = E0 [T0 | e0 = 1] = E0 [log(1 + SINR0) | e0 = 1] and also about its Laplace transform LT (s) = E0 [e−sT0 | e0 = 1] = E0 [(1 + SINR0)−s | e0 = 1] . 9 inria-00403040, version 4 - 4 Dec 2009
- 30. Let us now make the following simple observations: τ(r, λ1) = E0 [log(1 + SINR0) | e0 = 1] = ∞ Z 0 P0 { log(1 + SINR0) t | e0 = 1 } dt = ∞ Z 0 P0 { SINR0 et − 1 | e0 = 1 } dt = ∞ Z 0 pc(r, λ1, et − 1) dt = ∞ Z 0 pc(r, λ1, v) v + 1 dv (16.13) and similarly LT (s) = E0 [(1 + SINR0)−s | e0 = 1] = 1 − 1 Z 0 pc(r, λ1, t−1/s − 1) dt = 1 − s ∞ Z 0 pc(r, λ1, v) (1 + v)1+s dv , (16.14) provided P{SINR0 = T} = 0 for all T ≥ 0, which is true e.g. when F admits a density (cf. (16.3)). This, together with Propositions 16.2.2 and 16.2.4, lead to the following results: Corollary 16.2.6. Under the assumptions of Proposition 16.2.2 (namely for the model with Rayleigh fad- ing) and for the OPL 3 path-loss model (16.1) τ = β 2 ∞ Z 0 e−λ1K(β)r2v v β 2 −1 1 + v β 2 LW µ(Ar)β vβ/2 dv (16.15) and LT (s) = βs 2 ∞ Z 0 1 − e−λ1K(β)r2v LW µ(Ar)β vβ/2 v β 2 −1 1 + v β 2 1+s dv , (16.16) where K(β) is defined in (16.9). Corollary 16.2.7. Under the assumptions of Proposition 16.2.4 (namely in the model with general fading F), τ = ∞ Z 0 ∞ Z −∞ LI1 (2iπvsl(r)) LW (2iπvsl(r)) LF (−2iπs) − 1 2iπs(1 + v) dsdv (16.17) and LT (s) = 1 − s ∞ Z 0 ∞ Z −∞ LI1 (2iπvsl(r)) LW (2iπvsl(r)) LF (−2iπs) − 1 2iπs(1 + v)1+s dsdv . (16.18) Here is a direct application of the last results. 10 inria-00403040, version 4 - 4 Dec 2009
- 31. r = .25 r = .37 r = .5 r = .65 r = .75 r = .9 r = 1 Rayleigh 1.52 .886 .480 .250 .166 .0930 .0648 Erlang (8) 1.71 .942 .495 .242 .155 .0832 .0571 Table 16.1 Impact of the fading on the mean throughput τ for varying distance r. The Erlang distribution of order 8 mimics the no-fading case; we use OPL3 3 with A = 1, β = 4 and Rayleigh (exponential) thermal noise W with mean 0.01. Example 16.2.8. Table 16.1 shows how Rayleigh fading compares to the situation with no fading. The OPL 3 model is assumed with A = 1 and β = 4. We assume W to be exponential with mean 0.01. We use the formulas of the last corollaries; the Rayleigh case is with F exponential of parameter 1; to represent the no fading case within this framework we take F Erlang of high order (here 8) with the same mean 1 as the exponential. The reason for using Erlang rather than deterministic is that the latter does not satisfy the first technical condition of Proposition 16.2.4. We see that the presence of fading is beneficial in the far-field, and detrimental in the near-field. 16.2.4 Scaling Properties We show below that in the Poisson bipolar network model of Section 16.2.1, when using the OPL 3 model (16.1) and when W = 0, some interesting scaling properties can be derived. Denote by p̄c(r) = pc(r, 1, 1) the value of the probability of coverage calculated in this model with T ≡ 1, λ1 = 1, W ≡ 0 and with normalized virtual powers F̄j i = µFj i . Note that p̄c(r) does not depend on any parameter of the model other than the distribution of the normalized virtual power F̄. Proposition 16.2.9. In the Poisson bipolar network model of Section 16.2.1 with path-loss model (16.1) and W = 0 pc(r, λ1, T) = p̄c(rT1/β p λ1) . Proof. The Poisson point process Φ1 with intensity λ1 0 can be represented as {X0 i/ √ λ1 }, where Φ0 = {X0 i} is Poisson with intensity 1 (cf. Example 1.3.12 in Volume I). Because of this, under (16.1), the Poisson shot-noise interference variable I1 admits the following representation: I1 = λ β/2 1 I01 , where I01 is defined in the same manner as I1 but with respect to Φ0. Thus for W = 0, pc(r, λ1, T) = P(F ≥ T(Ar)β I1 ) = P µF ≥ µ(ArT1/β λ 1/2 1 )β I01 = p̄c(rT1/β p λ1 ) . Remark 16.2.10. The scaling of the coverage probability presented in Proposition 16.2.9 is the first of several examples of scaling in √ λ. Recall that this is also how the transport capacity scales in the well- known Gupta and Kumar law. There are three fundamental ingredients for obtaining this scaling in the present context: 11 inria-00403040, version 4 - 4 Dec 2009
- 32. • the scale invariance property of the Poisson p.p. (cf. Example 1.3.12 in Volume I), • the power-law form of OPL 3, • the fact that thermal noise was neglected. Note however the following important limitations concerning this scaling. First, when λ → ∞, the nodes are closer to each other and one may challenge the use of OPL 3 (the pole at the origin is not adequate for representing the path loss on small distances). On the other hand, when λ → 0, the transmission distances are very long; communications become noise limited and the assumption W = 0 may no longer be justified. 16.3 Spatial Performance Metrics When trying to maximize the coverage probability pc(r, λ1, T) or the throughput τ(r, λ1), one obtains de- generate maxima at r = 0. Assuming that our MANET features packets which have to reach some distant destination nodes, a more meaningful optimization consists in maximizing some distance-based character- istics. In the coverage scenario, we for instance consider the mean progress made in a typical transmission: prog(r, λ1, T) = rE0 [δ0] = rpc(r, λ1, T) . (16.19) Similarly, in the digital communication (or Shannon-throughput) scenario, we define the mean transport of a typical transmission as trans(r, λ1, T) = rE0 [T0] = rτ(r, λ1) . (16.20) These characteristics might still not lead to pertinent optimizations of the MANET, as they are concerned with one (typical) transmission. In particular, they are trivially maximized when p → 0, when transmissions are very efficient but very rare in the network. In fact, we need some network (social) performance metrics. Definition 16.3.1. We call • (spatial) density of successful transmissions, dsuc, the mean number of successful transmissions per surface unit; • (spatial) density of progress, dprog, the mean number of meters progressed by all transmissions taking place per surface unit; • (spatial) density of throughput, dthrou, the mean throughput per surface unit; • (spatial) density of transport, dtrans, the mean number of bit-meters transported per second and per unit of surface. The knowledge of pc(r, λ1, T) or τ(r, λ1) allows one to estimate these spatial network performance metrics. The link between individual and social characteristics is guaranteed by Campbell’s formula (cf. (2.9 in Volume I)). In what follows we make precise the meaning of the characteristics proposed in Definition 16.3.1 for Poisson bipolar MANETs. The density of successful transmissions can formally be seen as the mean number of successful trans- missions in some arbitrary subset B of the plane: dsuc(r, λ1, T) = 1 |B| E X i eiδi1(Xi ∈ B) . 12 inria-00403040, version 4 - 4 Dec 2009
- 33. By stationarity, the last quantity does not depend on the particular choice of set B. Let g(x, Φ̃) = 1(x ∈ B)e0δ0. The right hand side of the above equation can be expressed as 1 |B| E hZ R2 g(x, Φ̃ − x) Φ(dx) i , and by Campbell’s formula (2.9 in Volume I) is equal to λ |B| Z R2 E0 [g(x, Φ̃)] dx = λE0 [e0δ0] , which gives the following result dsuc(r, λ1, T) = λ1pc(r, λ1, T) = λppc(r, λp, T). (16.21) Similarly, the density of progress can be defined as the mean distance progressed by all transmissions taking place in some arbitrary subset B of the plane dprog(r, λ1, T) = 1 |B| E X i reiδi1(Xi ∈ B) = rλ1pc(r, λ1, T) , (16.22) by the same arguments as for (16.21). The spatial density of throughput is equal to dthrou(r, λ1) = 1 |B| E X i ei1(Xi ∈ B) log(1 + SINRi) = λ1τ(r, λ1) (16.23) and the density of transport to dtrans(r, λ1) = 1 |B| E X i eir1(Xi ∈ B) log(1 + SINRi) = λ1rτ(r, λ1). (16.24) In the following sections we focus on the optimization of the spatial performance of an Aloha MANET. 16.3.1 Optimization of the Density of Progress 16.3.1.1 Best MAP Given Some Transmission Distance We already mentioned that a good tuning of p should find a compromise between the average number of concurrent transmissions per unit area and the probability that a given authorized transmission is successful. To find such a compromise, one ought to maximize the density of progress, or equivalently the density of successful transmissions, dsuc(r, λp, T) = λp pc(r, λp, T), w.r.t. p, for a given r and λ. This can be done explicitly for the Poisson bipolar network model with Rayleigh fading. For this we first optimize w.r.t. λ assuming p fixed and then deduce from this the optimal MAP for some fixed λ. Define λmax = arg max 0≤λ∞ dsuc(r, λ, T) , whenever such a value of λ exists and is unique. The following result follows from Proposition 16.2.2. 13 inria-00403040, version 4 - 4 Dec 2009
- 34. Proposition 16.3.2. Under the assumptions of Proposition 16.2.2 (namely for the M W+M/M model), if p = 1, the unique maximum of the density of successful transmissions dsuc(r, λ, T) is attained at λmax = 2π ∞ Z 0 u 1 + l(u)/(Tl(r)) du −1 , and the maximal value is equal to dsuc(r, λmax, T) = e−1 λmax LW (µTl(r)) . In particular, assuming W ≡ 0 and OPL 3 model (16.1) λmax = 1 K(β)r2T2/β , (16.25) dsuc(r, λmax, T) = 1 eK(β)r2T2/β . (16.26) with K(β) defined in (16.9). Proof. The result follows from (16.7) by differentiation of the function λpc(r, λ, T) with respect to λ. The above result yields the following corollary concerning the tuning of the MAC parameter when λ is fixed. Corollary 16.3.3. Under assumptions of Proposition 16.2.2 with given r, the value of the MAP p that max- imizes the density of successful transmissions is pmax = min(1, λmax/λ) . In order to extend our observations to general fading (or equivalently to a general distribution for vir- tual power), let us assume that W = 0 and let us adopt model OPL 3. Then, for Rayleigh fading, using Lemma 16.2.9, we can easily show that λmax and dsuc(r, λmax) exhibit, up to some constant, the same de- pendence on the model parameters (namely r, T and µ) as that given in (16.25) and (16.26). Proposition 16.3.4. In the GI 0+M/GI bipolar model with OPL 3 and W = 0, λmax = const1 r2T2/β , and dsuc(r, λmax) = const2 r2T2/β , (16.27) where the constants const1 and const2 do not depend on r, T, µ, provided λmax is well defined. Proof. Assume that λmax is well defined. 3 By Lemma 16.2.9, const1 = arg maxλ≥0{λp̄c( √ λ)} and const2 = maxλ≥0{λp̄c( √ λ)}. 14 inria-00403040, version 4 - 4 Dec 2009
- 35. 0 0.005 0.01 0.015 0.02 0.025 0.03 0 0.05 0.1 0.15 0.2 Density of successful transmissions d suc MAP p Rayleigh Rician q=0.5 Rician q=0.9 Fig. 16.2 Density of successful transmissions dsuc for Aloha in function of p in the Rayleigh and the Rician (with q = 1/2 and q = .9) fading cases; λ = r = 1, W = 0, and T = 10dB; we use OPL 3 with A = 1 and β = 4. Example 16.3.5 (Rayleigh versus Rician fading). Figure 16.2 compares the density of success for Rayleigh and Rician fading. For this, we use the representations of Propositions 16.2.2 and Proposition 16.2.4, respectively. In the Rayleigh case, F is exponential with mean 1. In the Rician case F = q+(1−q)F0, with 0 ≤ q ≤ 1, where F0 is exponential with mean 1 and q represents the part of the energy received on the line-of-sight. The density of success is plotted in function of p. We again observe that higher variances are beneficial for high densities of transmitters (which is here equivalent to the far field case) and detrimental for low densities. However here, in each case, there is an optimal MAP, and when properly optimized, Spatial Aloha does better for lower variances (i.e. for Rician fading with higher q). 16.3.1.2 *General Definition of λmax In this section, we show that under some mild conditions, λmax is well defined and not degenerate (i.e. 0 λmax ∞) for a general GI W+M/GI model. Assume T 0. Note that dsuc(r, 0, T) = 0; so under some natural non-degeneracy assumptions, the maximum is certainly not attained at λ = 0. Proposition 16.3.6. Consider the GI W+M/GI bipolar model with p = 1 and general fading with a finite mean. Assume that l(r) 0 and is such that the generic SN I(λ) = P Xj∈e Φ Fj/l(|Xj|) admits a density for all λ 0. Then (1) If P{ F 0 } 0, then pc(r, x, T) (and so dsuc(r, x, T)) is continuous in x, so that the maxi- mum of the function x → dsuc(r, x, T) in the interval [0, λ] is attained for some 0 λmax ≤ λ; (2) If for all a 0, the following modified SN: I0 (λ) = X Xj∈e Φ 1(|Xj| a)Fj/l(|Xj|) 3The question of the definition of λmax is addressed in § 16.3.1.2. 15 inria-00403040, version 4 - 4 Dec 2009
- 36. has finite mean for all λ 0, then limx→∞ dsuc(r, x, T) = 0 and consequently, for sufficiently large λ, this maximum is attained for some λmax λ. The statement of the last proposition means that for a sufficiently large density of nodes λ, a nontrivial MAP 0 pmax 1 equal to pmax = λmax/λ optimizes the density of successful transmissions. Proof. (of Proposition 16.3.6) Recall that pc(r, λ, T) = P{ I(λ) ≤ F/(l(r)T−W) }, where the dependence of the SN variable I(λ) = I1 = I (note that p = 1) on the intensity of the Poisson p.p. was made explicit. By the thinning property of the Poisson p.p. (stated in Proposition 1.3.5 in Volume I) we can split the SN variable into two independent Poisson SN terms I(λ + ) = I(λ) + I(). Moreover, we can do this in such a way that I(), which is finite by assumption, almost surely converges to 0 when → 0. Consequently, 0 ≤ pc(r, λ, T) − pc(r, λ + , T) = P n F l(r)T − W − I() I(λ) ≤ F l(r)T − W o , and lim →0 pc(r, λ, T) − pc(r, λ + , T) = P n I(λ) = F l(r)T − W o = 0 , where the last equation is due to the fact that I(λ) is independent of F, W and admits a density. Splitting the Poisson p.p. of intensity λ into two Poisson p.p.s with intensity λ − and respectively, and considering the associated SN variables I(λ−) and I(), with I(λ) defined as their sum, one can show in a similar manner that lim→0 pc(r, λ − , T) − pc(r, λ, T) = 0. This concludes the proof of the first part of the proposition. We now prove the second part. Let G(s) = P{ F ≥ s }. Take 0 and such that E[I0(1)] = I 0 (1) ∞. By independence we have dsuc(r, λ, T) = λpc(r, λ, T) ≤ λE h P n F ≥ I0 (λ)Tl(r)|I0 (λ) oi ≤ J1 + J2 , where J1 = E h λ I0(λ) 1 I0 (λ) ≥ λ(I 0 (1) − ) I0 (λ)G I0 (λ)Tl(r) i J2 = λE h 1 I0 (λ) λ(I 0 (1) − ) i . Since E[F] = R ∞ 0 G(s) ds ∞, I0(λ)G I0(λ)Tl(r) is uniformly bounded in I0(λ) and I0(λ)G I0(λ)Tl(r) → 0 when I0(λ) → ∞. Moreover, one can construct a probability space such that I0(λ) → ∞ almost surely as λ → ∞. Thus, by Lebesgue’s dominated convergence theorem, we have limλ→∞ J1 = 0. For J2 and t 0 we have J2 ≤ λP0 { e−tI0(λ) ≥ e−λt(I 0 (1)−) ≤ λE0 e−tI0(λ)+λt(I 0 (1)−) ] = λ exp λ t(I 0 (1) − ) − 2π ∞ Z a s 1 − LF (t/l(s)) ds . Note that the derivative of t(I 0 (1) − ) − 2π R ∞ a s 1 − LF (t/l(s)) ds with respect to t at t = 0 is equal to I 0 (1) − − I 0 (1) 0. Thus, for some small t 0, J2 ≤ λe−λC for some constant C 0. This shows that limλ→∞ J2 = 0, which concludes the proof. 16 inria-00403040, version 4 - 4 Dec 2009
- 37. 16.3.1.3 Best Transmission Distance Given Some Transmitter Density Assume now some given intensity λ1 of transmitters. We look for the distance r which maximizes the mean density of progress, or equivalently the mean progress prog(r, λ1, T) = r pc(r, λ1, T). We denote by rmax = rmax(λ) = arg max r≥0 prog(r, λ, T) the best transmission distance for the density of transmitters λ whenever such a value exists and is unique. Let ρ = ρ(λ) = prog(rmax(λ), λ, T) be the optimal mean progress. Proposition 16.3.7. In the GI 0+M/GI bipolar model with OPL 3 function and W = 0, rmax(λ) = const3 T1/β √ λ , and ρ(λ) = const4 T1/β √ λ , (16.28) where the constants const3 and const4 do not depend on R, T, µ, provided rmax is well defined. If F is exponential (i.e. for Rayleigh fading) and l(r) given by (16.1) then const3 = 1/ p 2K(β) and const4 = 1/ p 2eK(β). Proof. The result for general fading follows from Lemma 16.2.9. The constants for the exponential case can be evaluated by (16.8). Remark 16.3.8. We see that the optimal distance rmax(λ) from transmitter to receiver is of the order of the distance to the nearest neighbor of the transmitter, namely 1/(2 √ λ), when λ → ∞. Notice also that for Rayleigh fading and l(r) given by (16.1) we have the general relation: 2r2 λmax(r) = rmax(λ)2 λ. (16.29) As before, one can show that for a general model, under some regularity conditions, prog(r, λ, T) is continuous in r and that the maximal mean progress is attained for some positive and finite r. We skip these technicalities. 16.3.1.4 Degeneracy of Two Step Optimization Assume for simplicity a W = 0 and OPL 3 path-loss (16.1). In Section 16.3.1.1, we found that for fixed r, the optimal density of successful transmissions dsuc is attained when the density of transmitters is equal to λ1 = λmax = const1/(r2T2/β). It is now natural to look for the distance r maximizing the mean progress for the network with this optimal density of transmitters. But by Proposition 16.3.4 sup r≥0 prog(r, λmax, T) = sup r≥0 r pc(r, λmax, T) = sup r≥0 r dsuc(r, λmax, T) λmax = sup r≥0 r const2 const1 = ∞ , 17 inria-00403040, version 4 - 4 Dec 2009
- 38. and thus the optimal choice of r consists in taking r = ∞, and consequently λmax = 0. From a practical point of view, this is not of course an acceptable answer. The fact that the optimal value is r = ∞ might be a consequence of the fact that we took W = 0. But even if this is the case, the above observation suggests that for a small W 0, a (possibly) finite optimal value of r would be too large from a practical point of view. In a network-perspective, one might better optimize a more “social” characteristic of the MANET like e.g. the density of progress dprog = λrpc(r, λ, T) first in λ and then in r. However in this case one obtains the opposite degenerate answer: sup r≥0 dprog(r, λmax, T) = sup r≥0 r dsuc(r, λmax, T) = sup r≥0 r const2 r2T2/β = 0 , which is attained for r = 0 and λmax = ∞. The above analysis shows that a better receiver model is needed to study the joint optimization in the transmission distance and in λ. We propose such models in Chapter 17, where the receivers are no longer sampled as independent marks of the Poisson p.p. of potential transmitters, but belong to the point process of potential transmitters; more precisely, they are chosen among the nodes which are silenced by Aloha during the considered time slot. As we shall see in Section 17.4 these degeneracies may then vanish. 16.3.2 Optimization of the Density of Transport 16.3.2.1 Best MAP Given Some Transmission Distance Define λtrans max = arg max 0≤λ∞ dtrans(r, λ) whenever such a value of λ exists and is unique. We have the following result. Proposition 16.3.9. In the M 0+M/M bipolar model with OPL 3 and W = 0, the unique maximum λtrans max of the density of transport dtrans(r, λ) is attained at λtrans max = x∗(β) r2K(β) , (16.30) where x∗(β) is the unique solution of the integral equation ∞ Z 0 e−xv v β 2 −1 1 + v β 2 dv = x ∞ Z 0 e−xv v β 2 1 + v β 2 dv . (16.31) Proof. One obtains this characterization by differentiating (16.15) w.r.t. λ1. Example 16.3.10. Consider the following model: r = 1, fading is Rayleigh with parameter µ = 1; at- tenuation is OPL 3 with A = 1 and β = 4. One finds a unique positive solution to (16.31) which gives λtrans max ≈ 0.157. The associated mean throughput per node is τ(r, λtrans max ) ≈ 0.898. If one defines T∗ by the Shannon-like formula τ(r, λtrans max ) = log(1 + T∗), one finds T∗ ≈ 0.8635, which is a much lower SINR target than what is usually retained within this setting. 18 inria-00403040, version 4 - 4 Dec 2009
- 39. 16.3.2.2 Best Transmission Distance Given a Density of Transmitters Assume now some given intensity λ1 = λp of transmitters. We look for the distance r which maximizes the mean density of transport, or equivalently the mean throughput τ(r, λ1). We denote by rtrans max = rtrans max (λ) = arg max r≥0 rτ(r, λ) the best transmission distance for this criterion, whenever such a value of r exists and is unique. Proposition 16.3.11. In the M 0+M/M bipolar model with OPL 3, the unique maximum of the density of trans- port dtrans(r, λ) is attained at rtrans max = s y∗(β) λK(β) , (16.32) where y∗(β) is the unique solution of the integral equation ∞ Z 0 e−yv v β 2 −1 1 + v β 2 dv = 2y ∞ Z 0 e−yv v β 2 1 + v β 2 dv . (16.33) Proof. One obtains this characterization by differentiating (16.15) w.r.t. r. We do not pursue this line of thought any further. Let us nevertheless point out that the last results can be extended to more general fading models and also that the same degeneracies as those mentioned above take place. 16.3.3 Spatial Reuse in Optimized Poisson MANETs In wireless networks, the MAC algorithm is supposed to prevent simultaneous neighboring transmissions from occurring, as often as possible, since such transmissions are likely to produce collisions. Some MAC protocols (as e.g. CSMA considered in Section 18.1) create exclusion zones to protect scheduled transmis- sions. Aloha creates a random exclusion disc around each transmitter. By this we mean that for an arbitrary radius there is some non-null probability that all the nodes in the disk with this radius do not transmit at a given time slot. Definition 16.3.12. We define the mean exclusion radius as the mean distance from a typical transmitter to its nearest concurrent transmitter Rexcl = E0 h min i6=0 {|Xi| : ei = 1} i . 19 inria-00403040, version 4 - 4 Dec 2009
- 40. Proposition 16.3.13. For the GI W+M/GI bipolar model, we have Rexcl = Rexcl(λ1) = 1 2 √ λ1 = 1 2 √ λp . (16.34) Proof. The probability that the distance from the origin to the nearest point in the Poisson p.p. Φ1 of intensity λ1 = λp is larger than s is equal to e−λ1πs2 (cf. Example 1.4.7 in Volume I). Thus we have Rexcl = R ∞ 0 e−λ1πs2 ds. Here are two questions pertaining to an optimized scenario and which can be answered using the results of the previous sections: • If r is given and p is optimized, how does the resulting Rexcl compare to r? • If λ is given and r is optimized, how does the resulting r compare to Rexcl? We address these questions in a unified way using a notion of spatial reuse analogous to the concept of spectral reuse used in cellular networks. Definition 16.3.14. The spatial reuse factor of the bipolar Aloha MANET is the ratio of the distance r between the transmitter and the receiver and the mean exclusion radius Rexcl. So if the spatial density of transmitters in this Aloha MANET is λ1, then Sreuse = Sreuse(λ1, r) = r Rexcl = 2r p λ1. (16.35) Here are a few illustrations. Example 16.3.15. Consider the GI W+M/GI bipolar model of Section 16.2.1. Assume that the path-loss model is OPL 3 and that W = 0. Assume the fixed coding scenario of Section 16.2.2 i.e., the success of a transmis- sion requires a SINR larger than or equal to T. We deduce from Proposition 16.3.4 that the spatial intensity of transmitters that maximizes the density of successful transmissions is λmax = const1/(r2T2/β). Hence by (16.34) Rexcl(λmax) = 1 2 √ λmax = r T1/β 2 √ const1 , (16.36) so that at the optimum, the spatial reuse Sreuse = 2 √ const1 T1/β , (16.37) is independent of r. For example, for β = 4 and Rayleigh fading, we can use the fact that const1 = 1/K(β) to evaluate the last expressions. For a SINR target of T = 10dB, Rexcl(λmax) ≈ 1.976r. Equivalently Sreuse ≈ 0.506. In order to have a spatial reuse larger than 1, one needs a SINR target less than 2 √ 2/π 4 = 0.657, that is less than -1.82 dB. 20 inria-00403040, version 4 - 4 Dec 2009
- 41. Example 16.3.16. Consider the Poisson bipolar network model of Section 16.2.1 with general fading, path- loss model (16.1) and W = 0 and target SINR T. Assume that the spatial density of transmitters is fixed and equal to λ. Let rmax(λ) denote the transmitter-receiver distance which maximizes the mean progress. We get from Proposition 16.3.7 that at the optimum r, Sreuse = const3 2T1/β , (16.38) for all values of λ. For β = 4 and Rayleigh fading, if we pick a SINR target of 10 dB, then Sreuse ≈ 0.358 only. Similarly, Sreuse 1 iff T (2/K(β))β/2. For β = 4, this is iff T 0.164 or equivalently T less than -7.84 dB. Example 16.3.17. Assume the Poisson bipolar network model of Section 16.2.1 with Rayleigh fading, path-loss model (16.1) and W = 0. Consider the optimal coding scenario with Shannon throughput of Section 16.2.3. The distance between transmitter and receiver is r. We deduce from Proposition 16.3.9 that in terms of density of transport, the best organization of the MANET is that where the spatial intensity of transmitters is λtrans max = x(β)/r2K(β). Hence, at the optimum, Rexcl = r 1 2 s K(β) x∗(β) , (16.39) so that Sreuse = 2 s x∗(β) K(β) , (16.40) a quantity that again does not depend on r. For β = 4, one gets x∗(β) ≈ 0.771, so that Rexcl ≈ 1.27r and Sreuse ≈ 0.790. Example 16.3.18. Consider the same scenario as in the last example but assume now that the intensity of transmitter is λ fixed. Proposition 16.3.11 shows that in order to maximize the mean throughput, in the optimal scenario, the transmitter-receivers distance rtrans max is such that Sreuse = 2 s y∗(β) K(β) . (16.41) For β = 4, y∗(β) ≈ 0.122 and Sreuse ≈ 0.314. 16.4 Opportunistic Aloha In the basic Spatial Aloha scheme, each node tosses a coin to access the medium independently of the fading variables. It is clear that something more clever can be done by combining the random selection of transmitters with the occurrence of good channel conditions. The general idea of opportunistic Aloha is to select the nodes with the channel fading larger than a certain threshold as transmitters in the reference time slot. This threshold may be deterministic or random (we assume fading variables to be observable which is needed for this scheme to be implementable; for more details on implementation issues see (Baccelli, Blaszczyszyn, and Mühlethaler 2009a)). 21 inria-00403040, version 4 - 4 Dec 2009
- 42. 16.4.1 Model Definition More precisely, in a Poisson MANET, opportunistic Aloha with random MAC threshold can be described by an i.m. Poisson p.p. e Φ = {(Xi, θi, yi, Fi)}, where {(Xi, yi, Fi)} is as described in items (1)–(4) on the enumerated list in Section 16.2, with item (2) replaced by: (2’) The medium access indicator ei of node i (ei = 1 if node i is allowed to transmit and 0 otherwise) is the following function of the virtual power Fi i : ei = 1(Fi i θi), where {θi} are new random i.i.d. marks, with a generic mark denoted by θ. Special cases of interest are – that where θ is constant, – that where θ is exponential with parameter ν. In this latter case one can obtain a closed-form expression for the coverage probability. We still assume that for each i, the components of (Fj i , j) are i.i.d. Note that {ei} are again i.i.d. marks of the point process e Φ (which of course depend on the marks {θi, Fi i }). In what follows, we also assume that for each i, the coordinates of (Fj i , j) are i.i.d. (cf. assumption (4) of the plain Aloha model of Section 16.2). The set of transmitters is hence a Poisson p.p. Φ1 (different from that in Section 16.2) with intensity λP(F θ) (where F is a typical Fi i and θ a typical θi, with (F, θ) independent). Thus in order to compare opportunistic Aloha to the plain Aloha described in Section 16.2, one can take p = P{ F θ }, where p is the MAP of plain Aloha, which guarantees the same density of (selected) transmitters at a given time slot. 16.4.2 Coverage Probability Note that the virtual power emitted by any node to its receiver, given it is selected by opportunistic Aloha (i.e. given ei = 1) has for law the distribution of F conditional on F θ. Below, we denote by Fθ a random variable with this law. By the independence of (Fj i , j), the virtual powers Fj i , j 6= i, toward other receivers are still distributed as F. Consequently, the interference I1 i experienced at any receiver has exactly the same distribution as in plain Aloha. Hence, the probability for a typical transmitter to cover its receiver can be expressed by the following three independent generic random variables b pc(r, λ1, T) = P{ Fθ Tl(r)(I1 + W) } , (16.42) where I1 is the generic SN generated by Poisson p.p. with intensity λ1 = P{F θ}λ and (non-conditioned) fading variables Fj (as in (16.3)). Using the Kendall-like notation of Section 5.3 in Volume I, we can see b pc(r, λ1, T) as the coverage probability in some GI1 W+M/GI2 model in which the distribution of interfering virtual powers (GI2) is different from that of the useful signal (GI1). 16.4.2.1 Rayleigh Fading and Exponential Threshold Case We begin our analysis of opportunistic Aloha by a comparison of b pc(r, λ1, T) and pc(r, λ1, T) of plain Aloha when all parameters (T, W, r, etc.) are the same. To get more insight we assume first Rayleigh fading. In this case, F is exponential with parameter µ and since θ is independent of F, by the lack of memory property of the exponential variable, given that F θ, the variables θ and F − θ are independent. Moreover, the conditional distribution of F − θ given F θ is also exponential with parameter µ. Denote 22 inria-00403040, version 4 - 4 Dec 2009
- 43. by e θ the conditional law of θ given that F θ. Consequently in the Rayleigh fading case (16.42) can be rewritten as b pc(r, λ1, T) = E e−µ(Tl(r)(I1+W)−e θ) + , (16.43) where I1, W, e θ are independent random variables with distributions described above and a+ = max(a, 0). Comparing (16.43) to the middle expression in (16.5) it is clear that the opportunistic scheme does better than plain Aloha with MAP p such that p = P(F θ) = E e−µθ = Lθ(µ). This follows from the fact that the intensity of transmitters is the same in both cases, which implies that the laws of interference coincide in both formulas. In order to evaluate how much better opportunistic Aloha does in the Rayleigh case, we now focus on the case when θ is exponential (of parameter ν). Note that in this case, e θ is also exponential of parameter µ + ν. This is thus a M1 W+M/M2 model. Proposition 16.4.1. Assume the Poisson bipolar network model of Section 16.2.1 with opportunistic Aloha MAC given by (2’) in Section 16.4.1. Assume Rayleigh fading (exponential F with parameter µ) and expo- nential distribution of θ with parameter ν. Then b pc(r, λ1, T) = µ + ν ν LI1 (µTl(r))LW (µTl(r)) − µ ν LI1 ((µ + ν)Tl(r))LW ((µ + ν)Tl(r)) , where LI1 is given by (16.6) with λ1 = λν/(µ + ν). If moreover W ≡ 0 and the OPL 3 model (16.1) is assumed, then b pc(r, λ1, ν) = µ + ν ν exp{−λ1T2/β r2 K(β)} − µ ν exp n −λ1 (µ + ν)T µ 2/β r2 K(β) o , with λ1 as above. Proof. Note that Φ1 is a Poisson p.p. of intensity λν/(ν + µ) and e θ is exponential of parameter µ + ν. Using (16.43) we have hence b pc(r, λ1, T) = E h Tl(r)(I1+W) Z 0 (µ + ν)e−(µ+ν)x e−µ(Tl(r)(I1+W)−x)dx i + E h ∞ Z Tl(r)(I1+W) (µ + ν)e−(µ+ν)x dx i = µ + ν ν E h e−µTl(r)(I1+W) − e−(µ+ν)Tl(r)(I1+W) i + E h e−(µ+ν)Tl(r)(I1+W) i , which completes the proof. Note that, as expected, when letting ν tend to infinity, under mild conditions, the first expression of the last proposition tends to LI1 (µTl(r))LW (µTl(r)), namely the formula (16.7) of plain Aloha with λ1 = λ or equivalently p = 1. 16.4.2.2 General Case For a general fading and distribution of θ (for instance deterministic θ, in which case obviously e θ = θ), the following result can be proved along the same lines as Proposition 16.2.4. 23 inria-00403040, version 4 - 4 Dec 2009
- 44. Proposition 16.4.2. Assume the Poisson bipolar network model of Section 16.2.1 with opportunistic Aloha MAC given by (2’) in Section 16.4.1 for some general distribution of F and θ. Take the same assumptions as in Proposition 16.2.4 except that the condition on F is replaced by the following one • E[Fθ] ∞ and Fθ admits a square integrable density. Then b pc(r, λ1, T) = ∞ Z −∞ LI1 (2iπl(r)Ts) LW (2iπl(r)Ts) LFθ (−2iπs) − 1 2iπs ds . (16.44) Proof. It follows immediately from Equation (16.42) and Corollary 12.2.2 in Volume I. In the case of Rayleigh fading and deterministic θ, the last theorem can be used since Fθ is then the convolution of a deterministic distribution (with mean θ) and an exponential one (with parameter µ); this satisfies the assumptions of the proposition. In this case, LFθ (s) = e−sθ µ µ+s . Obviously it can also be used when θ is exponentially distributed with intensity ν. In particular, for Rayleigh fading, we have LFθ (s) = µ+ν µ+ν+s µ µ+s. Example 16.4.3. Assume Rayleigh fading. In Figure 16.3 we plot the density of successful transmissions dsuc in function of the parameter ν for three different scenarios: (1) Opportunistic Aloha with a deterministic threshold θ with value 1/ν, where dsuc = λ1b pc(r, λ1, ν), with λ1 = λe−µ ν and b pc(r, λ1, ν) given by Proposition 16.4.2; (2) Opportunistic Aloha with a random exponential threshold with parameter ν, where dsuc = λ1b pc(r, λ1, ν), with λ1 = λν µ+ν and b pc(r, λ, ν) given by Proposition 16.4.1; (3) Plain Aloha where dsuc = 1/(eK(β)r2T2/β) is the optimal density of successful transmissions as obtained in Proposition 16.3.2 (this is of course a constant in ν). In the particular case considered in this figure, the density of transmitters covering their target receiver is approx. 56% larger in the optimal opportunistic scheme with exponential threshold than in plain Aloha and 134% larger in the deterministic case. It may look surprising that the curves for the random exponential and the deterministic threshold cases (1 and 2 above) differ so much. One should bear in mind the fact that the two associated MAPs are quite different: p = ν µ+ν in the former case and p = e−µ ν in the latter. Example 16.4.4 (Rayleigh versus Rician fading). (Example 16.3.5 continued). Figure 16.4 compares the density of success of opportunistic Aloha for Rayleigh and Rician (q = 1/2) fading. (The Rician case with q = .9 has thresholds θ larger than .9 and leads to very small densities of success; it is not displayed). The two curves are based on Proposition 16.4.2. Note first that for the two considered cases the density of transmitters are quite different: (exp(−θ) in the Rayleigh case and exp(−2θ) in the Rician case, for θ 1/2), which explain why the shapes of the curves are so different. Here, we see 24 inria-00403040, version 4 - 4 Dec 2009
- 45. 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.2 0.4 0.6 0.8 1 Density of successfull transmissions d suc ν 10 -3 x Opportunistic Aloha (constant θ = 1/ν) Opportunistic Aloha (exponential θ of rate ν) Plain Aloha Implementation of Opportunistic Aloha Fig. 16.3 The density of successful transmissions dsuc of opportunistic Aloha for various choices of θ. The propagation model is (16.1). We assume Rayleigh fading with mean 1, W = 0, λ = 0.001, T = 10dB, r = p 1/λ and β = 4. For comparison the constant value λmaxpc(r, λmax) of plain Aloha is plotted. 0 0.01 0.02 0.03 0.04 0.05 0.06 0 1 2 3 4 5 6 Density of successful transmissions d suc θ Rayleigh Rician q=0.5 Fig. 16.4 Density of successful transmissions dsuc for opportunistic Aloha in function of the deterministic threshold θ in the Rayleigh and Rician (with q = 1/2) fading cases; other parameters as in Figure 16.2. the opposite phenomenon compared to what was observed above: when properly optimized, opportunistic Aloha does better when the fading variance increases, namely does much better for Rayleigh fading than for Rician fading with q = 1/2. This is in fact quite natural since the aim of opportunistic Aloha is to leverage diversity: more fading diversity/variance is hence beneficial to this protocol when properly tuned. 16.4.3 Shannon Throughput We consider now the adaptive coding scenario and the Shannon throughput introduced in Section 16.2.3. The following result on the throughput of opportunistic Aloha is a corollary of Proposition 16.4.2 and for- 25 inria-00403040, version 4 - 4 Dec 2009
- 46. 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.2 0.4 0.6 0.8 1 Density of throughput d throu ν 10 -3 x Opportunistic Aloha (constant θ = 1/ν) Opportunistic Aloha (exponential θ of rate ν) Plain Aloha p=ν Fig. 16.5 The density of throughput of opportunistic Aloha as a function of its parameter ν and that of plain Aloha. Assumptions are as in Figure 16.3. mula (16.13). Corollary 16.4.5. Under the assumptions of Proposition 16.4.2, the mean Shannon throughput of the typical transmitting node is b τ = ∞ Z 0 ∞ Z −∞ LI1 (2iπvsl(r)) LW (2iπvsl(r)) LFθ (−2iπs) − 1 2iπs(1 + v) dsdv . (16.45) Since the assumptions of the last result hold in both deterministic and exponential θ case for Rayleigh fading, we can use (16.45) to evaluate the density of throughput in both cases. Example 16.4.6. Figure 16.5 plots the density of throughput dthrou for Rayleigh fading and the same three cases of θ as in Example 16.4.3. In the particular case considered in this figure, the density of throughput is approx. 48% larger in the optimal opportunistic scheme with exponential threshold than in plain Aloha and 93% larger in the deterministic case. Remark: The deterministic threshold case seems to always outperform the exponential threshold case when both are tuned optimally. 16.5 Conclusion The two main assumptions of the chapter are that time is slotted and that the receivers are at some predefined from their transmitters. The last assumption is relaxed in the next chapter. Let us conclude with a few comments on how to relax the slotted time assumption. Time–slots are required in e.g. TDMA (Time Division Multiple Access) and one of the well known ad- vantages of Aloha is that it does not require slotted time. In order to model non-slotted Aloha, one has to 26 inria-00403040, version 4 - 4 Dec 2009
- 47. take into account the fact that interference (and thus SINR) can vary during a given transmission as some other transmissions may start or terminate. A more detailed packet reception model is hence needed. For example, if one assumes a coding with sufficient interleaving, then one can consider that it is the aver- aged SINR, where the averaging is over the whole packet reception period, that determines the success of reception. In this case a mathematical analysis of non-slotted Aloha is possible e.g. along the lines pre- sented in (Błaszczyszyn and Radunović 2007; Błaszczyszyn and Radunović 2008); see also a forthcoming paper (Błaszczyszyn and Mühlethaler 2010). 27 inria-00403040, version 4 - 4 Dec 2009
- 49. 17 Receiver Selection in Spatial Aloha 17.1 Introduction In this chapter, we consider a few possible scenarios where the receiver of a given transmitter is not necessar- ily at distance r, as in the Poisson bipolar model (cf. Section 16.2.1) considered so far. In a MANET, some routing algorithm (cf. Section 25.3.1.2) specifies the receivers(s) (relay node(s)) of each given transmitter. The joint analysis and the joint design of MAC and routing are difficult tasks even if we assume the simplest MAC (Aloha). We shall come back to such cross-layer routing schemes in Chapter 22. In this chapter, we make a first step in this direction by proposing receiver models based on simplifying assumptions on the routing layer. Specifically, we assume one of the following routing principles: • Each transmitter selects its receiver as close by as possible. We saw in Section 16.3.1.4, in one of the two-step optimizations (with respect to λ and then r), that it is in some sense best for a transmitter to select his receiver as close by as possible. This justifies the class of routing models which is considered in Section 17.2 and which consist in making as small as possible hops. • Each transmitter targets all available (potential) receivers. This assumption, of interest in multi- cast scenarios, is considered in Section 17.3. • Each transmitter targets the most distant successful receiver. In this scenario, as in the previous one, each transmitter targets all available receivers; however, some selection policy is applied among the nodes which successfully receive the packet. One selects the node optimizing the packet progress in a given direction as the relay node. All other nodes discard the packet (this is not a broadcast scheme). This opportunistic scenario, in which no specific receiver is prescribed in advance by the routing algorithm, and where one selects the best hop at the given time step, is considered in Section 17.4. The above routing principles are considered in conjunction with the following models regarding receiver locations: • The independent receiver model, where the potential receivers form a stationary p.p. Φ0 with intensity λ0, independent of the transmitter p.p. Φ. This model, considered in Section 17.2.1, 29 inria-00403040, version 4 - 4 Dec 2009
- 50. corresponds to the situation where the nodes of Φ transmit to randomly located nodes (access points or relay stations) which are external to the MANET. Several further specifications of this model are of interest. For example: – The independent Poisson receiver model, where Φ0 is some homogeneous Poisson p.p. – The independent honeycomb receiver model, where Φ0 is some stationary hexagonal grid; cf. Example 4.2.5 in Volume I. – The independent Poisson + periodic receiver model, where Φ0 is the superposition of two independent p.p.s, a Poisson p.p. of intensity λ0 − 0, and a stationary grid (e.g. the hexagonal one) of intensity 0. The presence of the periodic stations provides an upper-bound on the distance to the nearest neighbor, which can be arbitrarily large in the pure Poisson receiver model. The advantages of this solution will be discussed in Section 17.5. • The MANET receiver model, where the transmitters of the MANET Φ choose their receivers in the original set Φ of nodes of the MANET. Two scenarios are considered in Section 17.2.2. – The nodes of Φ not allowed to access the shared medium form the set of potential re- ceivers; i.e., we have Φ0 = Φ0 = Φ Φ1. Note that this scenario assumes some kind of MAC-aware routing, since the pattern of actual receivers depends on the current MAC status of the nodes in MANET. – All the nodes of the MANET are considered as potential receivers; i.e. Φ0 = Φ. In this case each transmitter targets some nodes in the MANET without knowing their MAC status. This assumption requires additional specification of what happens if the picked receiver is also transmitting. The last section of the chapter (§ 17.5) is devoted to time–space scenarios and to the evaluation of the lo- cal delays, which are the random numbers of slots required to transmit a packet in the packet model. In most practical cases, these local delays are finite random variables but they have unexpected properties: in many cases, their mean value is infinite; in certain cases, they exhibit an interesting phase transition phenomenon which we propose to call the wireless contention phase transition and which has several incarnations. All this is central for the analysis of time–space routing in Part V. 17.2 Nearest Receiver Models Assume that the potential receivers form some stationary p.p. Φ0, which is either external to Φ, as in the independent receiver model, or a subset of Φ, as in the MANET receiver model. The common assumption of the present section is that each transmitter selects the nearest point of Φ0 as its receiver. Formally, this consists in replacing the assumption concerning the distribution of {yi} in (3) of the definition of the Poisson bipolar model of Section 16.2.1 by: (3’) The receiver yi of the transmitter Xi ∈ Φ is the point yi = Y ∗ i = arg minYi∈Φ0,Yi6=Xi {|Yi −Xi|}. In such a generic nearest receiver (NR) routing, we have to assume that the arg min is almost surely well defined. The (technical) condition Yi 6= Xi that the receiver is not located at the same place as its transmitter is automatically satisfied for all the examples considered in what follows, except the model of Section 17.2.2. 30 inria-00403040, version 4 - 4 Dec 2009
- 51. Note that e Φ0 = {Xi, ei, yi = Y ∗ i , Fi} is no longer an independently marked p.p., since the marks {Y ∗ i } jointly depend on Φ0. By specifying the joint distribution of Φ and Φ0, we have particular incarnations of this generic model. NR routing also requires some additional specifications on what happens if two or more transmitters pick the same receiver. Our analysis applies to the following two situations: either the receivers are capable of receiving more than one (in fact, an arbitrarily large) number of transmissions at the same time, or the target SINR T is such that T 1, which excludes such multiple receptions (cf. Remark 17.3.2). 17.2.1 Independent Receiver Models In the independent receiver model, the nearest receiver yi is almost surely well defined for all i (cf. Lemma 4.2.2 in Volume I). It is easy to calculate the probability of successful reception pc(INR, λ1, T) in this Independent Nearest Receiver (INR) model, provided one knows the distribution of the distance from the orign to the nearest point of Φ0. For example: Proposition 17.2.1. The coverage probability in the Poisson INR model of intensity λ0 is equal to pc(Poisson INR, λ1, T) = 2πλ0 ∞ Z 0 r exp(−λ0πr2 )pc(r, λp, T) dr , (17.1) where pc(r, λp, T) is the probability of coverage at distance r, evaluated for the Poisson bipolar model under the same assumptions except for the receiver location. Proof. One can easily evaluate the (tail of the) distribution function of the distance from the transmitter X0 = 0 to its receiver Y ∗ 0 under P0 (recall, P0 is the Palm probability associated to the p.p. Φ of the nodes in MANET; cf. Section 16.2.2) P0 { |Y ∗ 0 | r | e0 = 1 } = P0 { min Yi∈Φ0 {|Yi|} r | e0 = 1 } = P0 { Φ0{B0(r)} = 0 | e0 = 1 } = P{ Φ0{B0(r)} = 0 } = e−λ0πr2 , (17.2) where Bx(r) is the ball of radius r centered at x and where the last but one equality is due to the indepen- dence of Φ0 and Φ (cf. also Example 1.4.7 in Volume I). Thus the result follows when conditioning on |Y ∗ 0 | and using the independence of Φ0 and Φ. In the same vein, for the independent honeycomb receiver model, pc(Hex INR, λ1, T) = Z C pc(|x|, λp, T) dx , (17.3) where C is the hexagon centered at the origin of the plane of side length ∆ = q (2π √ 3)/λ0 (see Exam- ple 4.2.5 in Volume I). The other MANET performance metrics considered for the bipolar network model can be evaluated as well. The general formulas (16.13), (16.14) for the mean Shannon throughput and its Laplace transforms 31 inria-00403040, version 4 - 4 Dec 2009
- 52. remain true with the appropriate coverage probabilities. Similarly the expressions (16.21) and (16.23), for the spatial density of success and of throughput, remain valid. However, the evaluation of the mean progress and transport, as well as their densities have to be modified, due to the fact that the distance to the receiver is now a random variable. For example, for the Poisson INR model, we get the following corollary from the proof of Proposition 17.2.1. Proposition 17.2.2. In the Poisson INR model of intensity λ0, the mean progress and the mean transport are respectively equal to: prog(Poisson INR, λ1, T) = 2πλ0 ∞ Z 0 r2 exp(−λ0πr2 )pc(r, λp, T) dr , (17.4) and trans(Poisson INR, λ1) = 2πλ0 ∞ Z 0 r2 exp(−λ0πr2 )τ(r, λ1) dr . (17.5) With such modified mean progress and transport, the formulas for the spatial densities can be obtained by multiplication by λ1 (cf. formulas (16.22) and (16.24) for the original bipolar model). 17.2.2 MANET Receiver Models In the MANET receiver model, the following two situations aiming at making the smallest possible hops are of particular interest: • In the MANET Nearest Receiver (MNR) model, each transmitter picks the nearest node of Φ which is a receiver at the considered time slot as its next relay. • In the MANET Nearest Neighbor (MNN) model, we assume that each transmitter picks the nearest node of Φ (other than itself) as its receiver, regardless of whether the latter is authorized to transmit or not at the considered time slot. As already mentioned, this requires additional specification of what happens if (a) a node is chosen as the receiver of two or more different transmitters; (b) a node which is chosen as receiver by some transmitter is itself a transmitter (i.e. yi ∈ Φ1). For simplicity, in what follows we suppose that T 1, so that if (a) happens, then at most one of the transmitters is successful and the others fail (cf. Remark 17.3.2). If (b) happens we assume that the transmission fails.1 From the properties of independent thinnings of Poisson p.p. (cf. Proposition 1.3.5 in Volume I), for the Aloha MAC, at a given time slot, the transmitters Φ1 and the nodes which are not authorized to transmit, 1In practice the transmitted signal power exceeds the received signal power by at least 40dB, often about 100dB, so T would have to be extremely small for (b) to lead to a successful transmission. 32 inria-00403040, version 4 - 4 Dec 2009
- 53. denoted by Φ0, form two independent Poisson p.p.s. Thus, the probability of successful reception in the MNR model is equal to that for the Poisson INR model: pc(MNR, λ1, T) = pc(Poisson INR, λ1, T) , (17.6) with λ0 = (1 − p)λ. This extends to other characteristics (mean progress, mean transport, spatial densities) considered in Section 17.2.1. In order to evaluate the probability of successful reception in the MNN model, we also condition on the location of the nearest neighbor y0 = Y ∗ 0 . However this conditioning modifies the distribution of the interferers. Proposition 17.2.3. The coverage probability in the MNN model is equal to pc(MNN, λ1, T) = 2πλp(1 − p) ∞ Z 0 r exp(−λπr2 )p∗ c(r, λp, T) dr , where the conditional coverage probability p∗ c(r, λp, T), given the nearest node is at distance r and is a receiver, can be expressed using independent random variables F, W, I∗1(r) as p∗ c(r, λp, T) = P{ F ≥ Tl(r)(I∗1 (r) + W) } . In this formula, the SN I∗1(r) has for Laplace transform LI∗1(r)(s) = exp −λ1π ∞ Z 0 t 1 − LF (s/l(t)) dt + λ1 π/2 Z −π/2 ∞ Z 2r cos(θ) t 1 − LF (s/l(t)) dtdθ . (17.7) Proof. We condition on the location of the nearest neighbor y0 = Y ∗ 0 of X0 = 0 under P0. By Slivnyak’s theorem (cf. Theorem 1.4.5 in Volume I and also Remark 2.1.7 in Volume I), we know that, under P0, the nodes of Φ {X0} are distributed as those of the homogeneous Poisson p.p. Thus the distance |Y ∗ 0 − X0| = |Y ∗ 0 | has the same distribution as in the Poisson INR model with λ0 = λ; see (17.2). However, in the MNN model, given some particular location of y0 = Y ∗ 0 , one has to take the following fact into account: there are no MANET nodes (thus, in particular, no interferers) in B0(|y0|). Consequently, under P0, given Y ∗ 0 = y0, the value of I1 0 in (16.2) is no longer distributed as the generic SN I1 of Section 16.2.2, which was driven by the stationary Poisson p.p. of intensity λ1, but as the SN of Φ1 given that there are no nodes of Φ in B0(|y0|). Note that the location y0 at which we evaluate this last SN is on the boundary (and not in the center) of the empty ball. By the strong Markov property of Poisson p.p. (cf. Example 1.5.2 and Proposition 1.5.3 in Volume I), the distribution of a Poisson p.p. given that B0(|y0|) is empty is equal to the distribution of the (non-homogeneous) Poisson p.p. with intensity equal to 0 in B0(|y0|) and λ1 outside this ball. Putting these arguments together and exploiting the rotation invariance of the picture, we conclude the proof. Other characteristics can be evaluated in a way similar to what was done for the Poisson INR model in Section 17.2.1. In what follows we focus on the MNR model. In particular, we want to optimize the density of successful transmissions dsuc(MNR, λp, T) and the density of progress dprog(MNR, λp, T) w.r.t. the MAP p. Recall from Section 16.3.1.4, that the joint optimization of dprog(MNR, λp, T) in r and λ (or in p given λ) is degenerate in the Poisson bipolar model. For simplicity we consider only Rayleigh fading. The next result follows from (16.8): 33 inria-00403040, version 4 - 4 Dec 2009
- 54. Proposition 17.2.4. Consider the MNR model with Rayleigh fading, OPL 3 path loss model and no noise (W = 0). The density of successful transmission and the density of progress in this model are equal to dsuc(MNR, λp, T) = λp(1 − p) (1 − p) + pT2/βK(β)/π , (17.8) dprog(MNR, λp, T) = √ λp(1 − p) 2 (1 − p) + pT2/βK(β)/π 3/2 . (17.9) It is easy to see that both densities dsuc and dprog attain their maximal values for some 0 p 1 that does not depend on λ. For example the density of success admits the following optimal tuning of the parameter p. Corollary 17.2.5. Under the assumptions of Proposition 17.2.4, we have arg max 0≤p≤1 dsuc(MNR, λp, T) = 1 1 + T1/β p K(β)/π , max 0≤p≤1 dsuc(MNR, λp, T) = λ 1 + 1 T1/β √ K(β)/π 1 + T1/β p K(β)/π . 17.3 Multicast Mode In this section we consider the situation where each transmitter broadcasts some common data to many receiving nodes (see Chapter 25). More precisely, as in the MNR model considered above, we assume that the potential receivers Φ0 = Φ0 = Φ Φ1 are those nodes of the MANET Φ which are not authorized to transmit at the considered time slot (a similar analysis can be done for other receiver models) and that each transmitter of this time slot targets all the potential receivers of Φ0. The model features an i.m. p.p. Φ̃ = {Xi, ei, Fi} with Poisson nodes Φ = {Xi} and their MAC indica- tors {ei} as in Section 16.2.1, except that no prescribed receivers {yi} are considered; the virtual powers Fj i have the following modified interpretation: (4’) Fj i denotes the virtual power emitted by node i (provided ei = 1) towards node j in Φ0. 17.3.1 SINR-neighbors Let us consider the coverage scenario of Section 16.2.2 in which a successful transmission requires a SINR not smaller than some threshold T. More precisely, adapting Definition 16.2.1 to the present scenario, we say that Xj successfully captures the signal from Xi if SINRij = Fj i /l(|Xi − Xj|) W + I1 ij ≥ T , (17.10) where I1 ij = P Xk∈Φ1,k6=i,j Fj k /l(|Xk − Xj|). 34 inria-00403040, version 4 - 4 Dec 2009
- 55. Other documents randomly have different content
- 56. 20 juillet par M. d'Oubril. — La signature de ce traité décide lord Yarmouth à produire ses pouvoirs. — Lord Lauderdale est adjoint à lord Yarmouth. — Difficultés de la négociation avec l'Angleterre. — Quelques indiscrétions commises par les négociateurs anglais, au sujet de la restitution du Hanovre, font naître à Berlin de vives inquiétudes. — Faux rapports qui exaltent l'esprit de la cour de Prusse. — Nouvel entraînement des esprits à Berlin, et résolution d'armer. — Surprise et méfiance de Napoléon. — La Russie refuse de ratifier le traité signé par M. d'Oubril, et propose de nouvelles conditions. — Napoléon ne veut pas les admettre. — Tendance générale à la guerre. — Le roi de Prusse demande l'éloignement de l'armée française. — Napoléon répond par la demande d'éloigner l'année prussienne. — Silence prolongé de part et d'autre. — Les deux souverains partent pour l'armée. — La guerre est déclarée entre la Prusse et la France. 370 à 568 FIN DE LA TABLE DU SIXIÈME VOLUME.
- 57. Notes 1: Les Autrichiens n'ont jamais fait connaître leurs opérations dans cette première partie de la campagne de 1805. On a publié néanmoins beaucoup d'écrits en Allemagne, dans lesquels on s'est attaché à accabler le général Mack, à exalter l'archiduc Ferdinand, pour expliquer par l'ineptie d'un seul homme le désastre de l'armée autrichienne, et diminuer en même temps la gloire des Français. Ces écrits sont tous inexacts et injustes, et s'appuient la plupart du temps sur des circonstances fausses, dont l'impossibilité même est démontrée. Je me suis procuré avec beaucoup de peine l'un des rares exemplaires de la défense présentée par le général Mack au conseil de guerre devant lequel il fut appelé à comparaître. Cette défense, d'une forme singulière, d'un ton contraint, surtout à l'égard de l'archiduc Ferdinand, plus remplie de réflexions déclamatoires que de faits, m'a cependant fourni le moyen de bien préciser les intentions du général autrichien, et de rectifier un grand nombre de suppositions absurdes. Je crois donc être arrivé dans ce récit à la vérité, autant du moins qu'il est permis de l'espérer à l'égard d'événements qui n'ont pas été constatés par écrit même en Autriche, et qui sont presque sans témoins vivants aujourd'hui. Les principaux personnages en effet sont morts, et il y a eu en Allemagne un motif fort naturel, fort excusable de défigurer la vérité, celui de sauver l'amour-propre national en accablant un seul homme. 2: Voici l'énumération approximative, mais plutôt réduite qu'exagérée, de ces prisonniers: Pris à Wertingen 2,000
- 58. à Günzbourg 2,000 à Haslach 4,000 à Munich 1,000 à Elchingen 3,000 à Memmingen 5,000 Pendant la poursuite dirigée par Murat 12 à 13,000 Total 29 ou 30,000 3: On a fait une foule de conjectures sur les causes qui amenèrent la sortie en masse de la flotte de Cadix, et la bataille de Trafalgar. Il n'y a de vrai que ce que nous rapportons ici. Notre récit est emprunté à la correspondance authentique de Napoléon, et à celle des amiraux Decrès et Villeneuve. Il n'y a dans ce triste événement rien au delà de ce qu'on va lire. 4: C'est sur des pièces authentiques que je fonde cette assertion. 5: Au prince Murat. «Schœnbrunn, 25 brumaire an XIV (16 novembre 1805), à huit heures du matin. «Il m'est impossible de trouver des termes pour vous exprimer mon mécontentement. Vous ne commandez que mon avant- garde, et vous n'avez pas le droit de faire d'armistice sans mon ordre. Vous me faites perdre le fruit d'une campagne. Rompez l'armistice sur-le-champ et marchez à l'ennemi. Vous lui ferez déclarer que le général qui a signé cette capitulation n'avait point le droit de le faire; qu'il n'y a que l'empereur de Russie qui ait ce droit. «Toutes les fois, cependant, que l'empereur de Russie ratifierait ladite convention, je la ratifierais; mais ce n'est qu'une ruse; marchez, détruisez l'armée russe; vous êtes en position de
- 59. prendre ses bagages et son artillerie. L'aide de camp de l'empereur de Russie est un... Les officiers ne sont rien quand ils n'ont pas de pouvoirs: celui-ci n'en avait point. Les Autrichiens se sont laissé jouer pour le passage du pont de Vienne, vous vous laissez jouer par un aide de camp de l'empereur...» 6: Les Russes l'ont portée à beaucoup moins le lendemain de leur défaite, Napoléon à beaucoup plus dans ses bulletins. Après la confrontation d'un grand nombre de témoignages et d'états authentiques, nous croyons présenter ici l'assertion la plus exacte. 7: Il vient de paraître un écrit traduit du russe par M. Léon de Narischkine, lequel contient un grand nombre d'assertions inexactes, quoique publié par un auteur en position d'être bien informé. Dans cet écrit il est dit que Napoléon eut avant la bataille d'Austerlitz communication du plan du général Weirother. Cette allégation est tout à fait erronée. Une pareille communication ne serait explicable que si le plan, communiqué longtemps d'avance aux divers chefs de corps, avait pu être exposé à une divulgation. On verra ci-après, par le rapport d'un témoin oculaire, que c'est seulement dans la nuit qui précéda la bataille, que le plan fut communiqué aux chefs de corps. Du reste, tous les détails des ordres et de la correspondance prouvent que Napoléon prévit et ne connut pas le plan de l'ennemi. Notre résolution étant d'éviter toute polémique avec les auteurs contemporains, nous nous bornerons à redresser cette erreur, sans nous occuper de beaucoup d'autres, que renferme encore l'ouvrage en question, dont nous reconnaissons d'ailleurs le mérite très-réel, et jusqu'à un certain point l'impartialité. 8: Nous croyons utile de citer un fragment des mémoires manuscrits du général Langeron, témoin oculaire, puisqu'il commandait l'un des corps de l'armée russe. Voici le récit de cet officier:
- 60. «On a vu que, le 19 novembre (1er décembre), nos colonnes ne parvinrent à leur destination que vers les dix heures du soir. »Vers les onze heures, tous les chefs de ces colonnes, excepté le prince Bagration, qui était trop éloigné, reçurent l'ordre de se rendre à Kreznowitz, chez le général Kutusof, afin d'entendre la lecture des dispositions pour la bataille du lendemain. »À une heure du matin, lorsque nous fûmes tous rassemblés, le général Weirother arriva, déploya sur une grande table une immense carte très-exacte des environs de Brünn et d'Austerlitz, et nous lut ses dispositions, d'un ton élevé et avec un air de jactance qui annonçaient en lui la persuasion intime de son mérite et celle de notre incapacité. Il ressemblait à un régent de collége qui lit une leçon à de jeunes écoliers. Nous étions peut- être effectivement des écoliers; mais il était loin d'être un bon professeur. Kutusof, assis et à moitié endormi lorsque nous arrivâmes chez lui, finit par s'endormir tout à fait avant notre départ. Buxhoewden, debout, écoutait, et sûrement ne comprenait rien; Miloradovitch se taisait; Pribyschewski se tenait en arrière, et Doctoroff seul examinait la carte avec attention. Lorsque Weirother eut fini de pérorer, je fus le seul qui pris la parole. Je lui dis: «Mon général, tout cela est fort bien; mais si les ennemis nous préviennent et nous attaquent près de Pratzen, que ferons-nous?»—«Le cas n'est pas prévu, me répondit-il; vous connaissez l'audace de Buonaparte. S'il eût pu nous attaquer, il l'eût fait aujourd'hui.»—«Vous ne le croyez donc pas fort? lui dis- je.»—«C'est beaucoup s'il a 40,000 hommes.»—«Dans ce cas, il court à sa perte en attendant notre attaque; mais je le crois trop habile pour être imprudent, car si, comme vous le voulez et le croyez, nous le coupons de Vienne, il n'a d'autre retraite que les montagnes de la Bohême; mais je lui suppose un autre projet. Il a éteint ses feux, on entend beaucoup de bruit dans son camp.»—«C'est qu'il se retire ou qu'il change de position; et même, en supposant qu'il prenne celle de Turas, il nous épargne beaucoup de peine, et les dispositions restent les mêmes.»
- 61. »Kutusof alors, s'étant réveillé, nous congédia en nous ordonnant de laisser un adjudant pour copier les dispositions que le lieutenant colonel Toll, de l'état-major, allait traduire de l'allemand en russe. Il était alors près de trois heures du matin, et nous ne reçûmes les copies de ces fameuses dispositions qu'à près de huit heures, lorsque déjà nous étions en marche.» 9: Le prince Czartoryski, placé entre les deux empereurs, fit remarquer à l'empereur Alexandre la marche leste et décidée des Français qui gravissaient le plateau, sans répondre au feu des Russes. Ce prince ému à cette vue sentit défaillir la confiance qu'il avait éprouvée jusque-là, et en conçut un pressentiment sinistre qui ne l'abandonna pas de la journée. 10: Celui qui est mort récemment. 11: Les nombres exacts n'étaient pas encore connus. 12: C'est M. de Talleyrand qui raconte ce détail dans une de ses lettres à Napoléon. 13: J'emprunte ce récit aux sources les plus authentiques: aux Mémoires du prince Cambacérès d'abord, puis aux Mémoires intéressants et instructifs de M. le comte Mollien, qui ne sont point encore publiés, et enfin aux Archives du Trésor. J'ai tenu et lu moi- même, avec une grande attention, les pièces du procès, et surtout un long et intéressant rapport que le ministre du Trésor rédigea pour l'Empereur. Je n'avance donc rien ici que sur preuves officielles et incontestables. 14: Nous citons la lettre suivante, qui reproduit exactement la pensée de Napoléon dans cette circonstance: À M. de Talleyrand. Paris, 4 février 1806.
- 62. Le ministère en Angleterre a été entièrement changé après la mort de M. Pitt. M. Fox a le portefeuille des relations extérieures. Je désire que vous me présentiez ce soir une note rédigée sur cette idée: «Le soussigné ministre des relations extérieures a reçu l'ordre exprès de S. M. l'Empereur de faire connaître à M. d'Haugwitz, à sa première entrevue, que S. M. ne saurait regarder le traité conclu à Vienne comme existant, par défaut de ratification dans le temps prescrit; que S. M. ne reconnaît à aucune puissance, et moins à la Prusse qu'à toute autre, parce que l'expérience a prouvé qu'il faut parler clairement et sans détour, le droit de modifier et d'interpréter, selon son intérêt, les différents articles d'un traité; que ce n'est pas échanger des ratifications que d'avoir deux textes différents d'un même traité, et que l'irrégularité paraît encore plus grande si l'on considère les trois ou quatre pages de mémoire ajoutées aux ratifications de la Prusse; que M. de Laforest, ministre de S. M., chargé de l'échange des ratifications, serait coupable, si lui-même n'eût observé toute l'irrégularité du procédé de la cour de Prusse, mais qu'il n'avait accepté l'échange qu'avec la condition de l'approbation de l'Empereur. »Le soussigné est donc chargé de déclarer que S. M. ne l'approuve pas, par la considération de la sainteté due à l'exécution des traités. »Mais en même temps le soussigné est chargé de déclarer que S. M. désire toujours que les différends survenus dans ces dernières circonstances entre la France et la Prusse se terminent à l'amiable, et que l'ancienne amitié qui avait existé entre elles subsiste comme par le passé; elle désire même que le traité d'alliance offensive et défensive, s'il est compatible avec les autres engagements de la Prusse, subsiste entre les deux pays et assure leurs liaisons.»
- 63. Cette note, que vous me présenterez ce soir, sera remise demain dans la conférence, et sous quelque prétexte que ce soit je ne vous laisse pas le maître de ne la pas remettre. Vous comprenez vous-même que ceci a deux buts: de me laisser maître de faire ma paix avec l'Angleterre, si d'ici à quelques jours les nouvelles que je reçois se confirment, ou de conclure avec la Prusse un traité sur une base plus large. Vous serez sévère et net dans la rédaction; mais vous y ajouterez de vive voix toutes les modifications, tous les adoucissements, toutes les illusions qui feront croire à M. d'Haugwitz que c'est une suite de mon caractère, qui est piqué de cette forme, mais que dans le fond on est dans les mêmes sentiments pour la Prusse. Mon opinion est que dans les circonstances actuelles, si véritablement M. Fox est à la tête des affaires étrangères, nous ne pouvons céder le Hanovre à la Prusse que par suite d'un grand système tel qu'il puisse nous garantir de la crainte d'une continuation d'hostilités. 15: Texte de la dépêche. 16: Nous citons le curieux document qui fut adressé à Napoléon. Ratisbonne, 19 avril 1806. Sire, Le génie de Napoléon ne se borne pas à créer le bonheur de la France; la Providence accorde l'homme supérieur à l'univers. L'estimable nation germanique gémit dans les malheurs de l'anarchie politique et religieuse: soyez, Sire, le régénérateur de sa Constitution! Voici quelques vœux dictés par l'état des choses. Que le duc de Clèves devienne électeur, qu'il obtienne l'octroi du Rhin sur toute la rive droite; que le cardinal Fesch soit mon coadjuteur; que les rentes assignées sur l'octroi à douze États de l'empire soient fondées sur quelque autre base. Votre Majesté Impériale et Royale jugera dans sa sublimité s'il est utile au bien
- 64. général de réaliser ces idées. Si quelque erreur idéologique me trompe à cet égard, le cœur m'atteste au moins la pureté de mes intentions. Je suis avec un attachement inviolable et le plus profond respect, Sire, de Votre Majesté Impériale et Royale le très-humble et tout dévoué admirateur, Charles, électeur archichancelier. La nation germanique a besoin que sa Constitution soit régénérée: la majeure partie de ses lois ne présente que des mots vides de sens, depuis que les tribunaux, les cercles, la Diète de l'empire n'ont plus les moyens nécessaires pour soutenir les droits de propriété et de sûreté personnelle des individus qui composent la nation, et que ces institutions ne peuvent plus protéger les opprimés contre les attentats du pouvoir arbitraire et de la cupidité. Un tel état est anarchique; les peuples supportent les charges de l'état civil sans jouir de ses principaux avantages, position désastreuse pour une nation foncièrement estimable par sa loyauté, son industrie, son énergie primitive. La Constitution germanique ne peut être régénérée que par un chef de l'empire d'un grand caractère, qui rende la vigueur aux lois en concentrant dans ses mains le pouvoir exécutif. Les États de l'empire n'en jouiront que d'autant mieux de leurs domaines, lorsque les vœux des peuples seront exposés et discutés à la Diète, les tribunaux mieux organisés, et la justice administrée d'une manière plus efficace. Sa Majesté l'empereur d'Autriche, François second, serait un particulier respectable par ses qualités personnelles, mais dans le fait le sceptre d'Allemagne lui échappe, parce qu'il a maintenant la majorité de la Diète contre lui; qu'il a manqué à sa capitulation en occupant la Bavière, en introduisant les Russes en Allemagne, en démembrant des parties de l'empire pour payer des fautes commises dans les querelles particulières de sa maison. Puisse-t-il être empereur
- 65. d'Orient pour résister aux Russes, et que l'empire d'Occident renaisse en l'empereur Napoléon, tel qu'il était sous Charlemagne, composé de l'Italie, de la France et de l'Allemagne! Il ne paraît pas impossible que les maux de l'anarchie fassent sentir la nécessité d'une telle régénération à la majorité des électeurs; c'est ainsi qu'ils choisirent Rodolphe de Habsbourg après les troubles du grand interrègne. Les moyens de l'archichancelier sont très-bornés; mais c'est au moins avec une intention pure qu'il compte sur les lumières de l'empereur Napoléon, nommément dans les objets qui pourront agiter le midi de l'Allemagne plus particulièrement dévoué à ce monarque. La régénération de la Constitution germanique a été de tout temps l'objet des vœux de l'électeur archichancelier; il ne demande et n'accepterait rien pour lui-même; il pense que si Sa Majesté l'empereur Napoléon pouvait se réunir en personne chaque année pour quelques semaines à Mayence ou ailleurs avec les princes qui lui sont attachés, les germes de la régénération germanique se développeraient bientôt. M. d'Hédouville a inspiré une parfaite confiance à l'électeur archichancelier, qui sera charmé s'il veut bien exposer ces idées dans toute leur pureté à Sa Majesté l'empereur des Français et à son ministre M. de Talleyrand. Charles, électeur archichancelier. 17: C'est de M. de Labesnardière lui-même, seul confident de cette importante création, que nous tenons tous ces détails, appuyés en outre sur une foule de documents authentiques. 18: Nous citons les lettres suivantes, qui montrent comment Napoléon donnait les couronnes et comment on les recevait. «Au ministre de la guerre. Munich, 5 janvier 1806.
- 66. «Expédiez le général Berthier, votre frère, avec le décret qui nomme le prince Joseph commandant de l'armée de Naples. Il gardera le plus profond secret, et ce ne sera que lorsque le prince arrivera qu'il lui remettra le décret. Je dis qu'il doit garder le plus profond secret, parce que je ne suis pas sûr que le prince Joseph y aille, et, sous ce point, il ne faut pas que rien soit connu.» «Au prince Joseph. »Stuttgard, le 19 janvier 1806. »Mon intention est que dans les premiers jours de février vous entriez dans le royaume de Naples, et que je sois instruit dans le courant de février que mes aigles flottent sur cette capitale. Vous ne ferez aucune suspension d'armes ni capitulation. Mon intention est que les Bourbons aient cessé de régner à Naples, et je veux sur ce trône asseoir un prince de ma maison, vous d'abord, si cela vous convient, un autre si cela ne vous convient point. »Je vous réitère de ne point diviser vos forces; que toute votre armée passe l'Apennin, et que vos trois corps d'armée soient dirigés droit sur Naples, de manière à se réunir en un jour sur un même champ de bataille. »Laissez un général, des dépôts, des approvisionnements et quelques canonniers à Ancône pour défendre la place. Naples pris, les extrémités tomberont d'elles-mêmes, tout ce qui sera dans les Abbruzzes sera pris à revers, et vous enverrez une division à Tarente, et une du côté de la Sicile pour achever la conquête du royaume. »Mon intention est de laisser sous vos ordres dans le royaume de Naples pendant l'année, jusqu'à ce que j'aie fait de nouvelles dispositions, 14 régiments d'infanterie française, complétés au
- 67. grand complet de guerre, et 12 de cavalerie française aussi au grand complet. »Le pays doit vous fournir les vivres, l'habillement, les remontes, et tout ce qui est nécessaire, de manière qu'il ne m'en coûte pas un sou. Mes troupes du royaume d'Italie n'y resteront qu'autant de temps que vous le jugerez nécessaire, après quoi elles retourneront chez elles. »Vous lèverez une légion napolitaine où vous ne laisserez entrer que des officiers et soldats napolitains, des gens du pays qui voudront s'attacher à ma cause.» 19: J'ai lu toutes ces dépêches avec la plus grande attention; et comme je dis la vérité à l'égard de toutes les cours, grandes et petites, je la dirais à l'égard de la Hesse, cette vérité lui fût-elle favorable, et fût-elle défavorable à la France. 20: Cette lettre existe au dépôt de la Secrétairerie d'État au Louvre.
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